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+# FjORD: Fair and Accurate Federated Learning under heterogeneous targets with Ordered Dropout
+
+Samuel Horváth∗ KAUST† Thuwal, KSA samuel.horvath@kaust.edu.sa
+
+Stefanos Laskaridis∗ Samsung AI Center Cambridge, UK mail@stefanos.cc
+
+Mario Almeida∗   
+Samsung AI Center   
+Cambridge, UK   
+mario.a@samsung.com
+
+Ilias Leontiadis Samsung AI Center Cambridge, UK i.leontiadis@samsung.com
+
+Stylianos I. Venieris Samsung AI Center Cambridge, UK s.venieris@samsung.com
+
+Nicholas D. Lane Samsung AI Center &   
+University of Cambridge Cambridge, UK   
+nic.lane@samsung.com
+
+# Abstract
+
+Federated Learning (FL) has been gaining significant traction across different ML tasks, ranging from vision to keyboard predictions. In large-scale deployments, client heterogeneity is a fact and constitutes a primary problem for fairness, training performance and accuracy. Although significant efforts have been made into tackling statistical data heterogeneity, the diversity in the processing capabilities and network bandwidth of clients, termed as system heterogeneity, has remained largely unexplored. Current solutions either disregard a large portion of available devices or set a uniform limit on the model’s capacity, restricted by the least capable participants. In this work, we introduce Ordered Dropout, a mechanism that achieves an ordered, nested representation of knowledge in deep neural networks (DNNs) and enables the extraction of lower footprint submodels without the need of retraining. We further show that for linear maps our Ordered Dropout is equivalent to SVD. We employ this technique, along with a self-distillation methodology, in the realm of FL in a framework called FjORD. FjORD alleviates the problem of client system heterogeneity by tailoring the model width to the client’s capabilities. Extensive evaluation on both CNNs and RNNs across diverse modalities shows that FjORD consistently leads to significant performance gains over state-of-the-art baselines, while maintaining its nested structure.
+
+# 1 Introduction
+
+Over the past few years, advances in deep learning have revolutionised the way we interact with everyday devices. Much of this success relies on the availability of large-scale training infrastructures and the collection of vast amounts of training data. However, users and providers are becoming increasingly aware of the privacy implications of this ever-increasing data collection, leading to the creation of various privacy-preserving initiatives by service providers [3] and government regulators [10].
+
+Federated Learning (FL) [46] is a relatively new subfield of machine learning (ML) that allows the training of models without the data leaving the users’ devices; instead, FL allows users to collaboratively train a model by moving the computation to them. At each round, participating devices download the latest model and compute an updated model using their local data. These locally trained models are then sent from the participating devices back to a central server where updates are aggregated for next round’s global model. Until now, a lot of research effort has been invested with the sole goal of maximising the accuracy of the global model [46, 42, 39, 31, 63], while complementary mechanisms have been proposed to ensure privacy and robustness [6, 14, 47, 48, 27, 4].
+
+![](images/2e8b497a9c0ca5985656f9912ce45ced32adba2df264ad6a401b94f9458816d5.jpg)  
+Figure 1: FjORD employs OD to tailor the amount of computation to the capabilities of each participating device.
+
+![](images/b24e2607785994e09540f47c635a573b80e105dcd201e1a9e45ad3e10f6a629c.jpg)  
+Figure 2: Ordered vs. Random Dropout. In this example, the left-most features are used by more devices during training, creating a natural ordering to the importance of these features.
+
+A key challenge of deploying FL in the wild is the vast heterogeneity of devices [38], ranging from low-end IoT to flagship mobile devices. Despite this fact, the widely accepted norm in FL is that the local models have to share the same architecture as the global model. Under this assumption, developers typically opt to either drop low-tier devices from training, hence introducing training bias due to unseen data [30], or limit the global model’s size to accommodate the slowest clients, leading to degraded accuracy due to the restricted model capacity [8]. In addition to these limitations, variability in sample sizes, computation load and data transmission speeds further contribute to a very unbalanced training environment. Finally, the resulting model might not be as efficient as models specifically tailored to the capabilities of each device tier to meet the minimum processingperformance requirements [34].
+
+In this work, we introduce FjORD (Fig. 1), a novel adaptive training framework that enables heterogeneous devices to participate in FL by dynamically adapting model size – and thus computation, memory and data exchange sizes – to the available client resources. To this end, we introduce Ordered Dropout (OD), a mechanism for run-time ordered (importance-based) pruning, which enables us to extract and train submodels in a nested manner. As such, OD enables all devices to participate in the FL process independently of their capabilities by training a submodel of the original DNN, while still contributing knowledge to the global model. Alongside OD, we propose a self-distillation method from the maximal supported submodel on a device to enhance the feature extraction of smaller submodels. Finally, our framework has the additional benefit of producing models that can be dynamically scaled during inference, based on the hardware and load constraints of the device.
+
+Our evaluation shows that FjORD enables significant accuracy benefits over the baselines across diverse datasets and networks, while allowing for the extraction of submodels of varying FLOPs and sizes without the need for retraining.
+
+# 2 Motivation
+
+Despite the progress on the accuracy front, the unique deployment challenges of FL still set a limit to the attainable performance. FL is typically deployed on either siloed setups, such as among hospitals, or on mobile devices in the wild [7]. In this work, we focus on the latter setting. Hence, while cloud-based distributed training uses powerful high-end clients [19], in FL these are commonly substituted by resource-constrained and heterogeneous embedded devices.
+
+In this respect, FL deployment is currently hindered by the vast heterogeneity of client hardware [66, 28, 7]. On the one hand, different mobile hardware leads to significantly varying processing speed [1], in turn leading to longer waits upon aggregation of updates (i.e. stragglers). At the same time, devices of mid and low tiers might not even be able to support larger models, e.g. the model does not fit in memory or processing is slow, and, thus, are either excluded or dropped upon timeouts from the training process, together with their unique data. More interestingly, the resource allocation to participating devices may also reflect on demographic and socio-economic information of owners, that makes the exclusion of such clients unfair [30] in terms of participation. Analogous to the device load and heterogeneity, a similar trend can be traced in the downstream (model) and upstream (updates) network communication in FL, which can be an additional substantial bottleneck for the training procedure [55].
+
+# 3 Ordered Dropout
+
+In this paper, we firstly introduce the tools that act as enablers for heterogeneous federated training. Concretely, we have devised a mechanism of importance-based pruning for the easy extraction of subnetworks from the original, specially trained model, each with a different computational and memory footprint. We name this technique Ordered Dropout (OD), as it orders knowledge representation in nested submodels of the original network.
+
+More specifically, our technique starts by sampling a value (denoted by $p$ ) from a distribution of candidate values. Each of these values corresponds to a specific submodel, which in turn gets translated to a specific computational and memory footprint (see Table 1b). Such sampled values and associations are depicted in Fig. 2. Contrary to conventional dropout (RD), our technique drops adjacent components of the model instead of random neurons, which translates to computational benefits3 in today’s linear algebra libraries and higher accuracy as shown later.
+
+# 3.1 Ordered Dropout Mechanics
+
+The proposed OD method is parametrised with respect to: i) the value of the dropout rate $p \in ( 0 , 1 ]$ per layer, ii) the set of candidate values $\mathcal { P }$ , such that $p \in \mathcal P$ and iii) the sampling method of $p$ over the set of candidate values, such that $p \sim D _ { \mathcal { P } }$ , where $D _ { \mathcal { P } }$ is the distribution over $\mathcal { P }$ .
+
+A primary hyperparameter of OD is the dropout rate $p$ which defines how much of each layer is to be included, with the rest of the units dropped in a structured and ordered manner. The value of $p$ is selected by sampling from the dropout distribution $D _ { \mathcal { P } }$ which is represented by a set of discrete values $\mathcal { P } = \{ s _ { 1 } , s _ { 2 } , . . . , s _ { | \mathcal { P } | } \}$ such that $0 { < } s _ { 1 } { < } . . . { < } s _ { | } p _ { | } \leq 1$ and probabilities $\mathbf { P } ( { \dot { p } } = s _ { i } ) > 0$ , $\forall i \in [ | \mathcal { P } | ]$ such that $\begin{array} { r } { \sum _ { i = 1 } ^ { | \mathcal { P } | } \mathbf { P } ( p = s _ { i } ) = 1 } \end{array}$ . For instance, a uniform distribution over $\mathcal { P }$ is denoted by $p \sim \mathcal { U } _ { \mathcal { P } }$ (i.e. $D = \mathcal { U }$ ). In our experiments we use uniform distribution over the set $\mathcal { P } = \{ i /  { k } \} _ { i = 1 } ^ { k }$ , which we refer to as $\mathcal { U } _ { k }$ (or uniform- $k$ ). The discrete nature of the distribution stems from the innately discrete number of neurons or filters to be selected. The selection of set $\mathcal { P }$ is discussed in the next subsection.
+
+The dropout rate $p$ can be constant across all layers or configured individually per layer $l$ , leading to $p _ { l } \sim \dot { D } _ { \mathcal { P } } ^ { l }$ . As such an approach opens the search space dramatically, we refer the reader to NAS techniques [69] and continue with the same $p$ value across network layers for simplicity, without hurting the generality of our approach.
+
+Given a $p$ value, a pruned $p$ -subnetwork can be directly obtained as follows. For each4 layer $l$ with width5 $K _ { l }$ , the submodel for a given $p$ has all neurons/filters with index $\{ 0 , 1 , \ldots , \lceil p \cdot K _ { l } \rceil - 1 \}$ included and $\{ \lceil p \cdot K _ { l } \rceil , \ldots , K _ { l } ^ { - } - 1 \}$ pruned. Moreover, the unnecessary connections between pruned neurons/filters are also removed6. We denote a pruned $p$ -subnetwork $\mathbf { F } _ { p }$ with its weights ${ \pmb w } _ { p }$ , where $\mathbf { F }$ and $\pmb { w }$ are the original network and weights, respectively. Importantly, contrary to existing pruning techniques [18, 35, 49], a $p$ -subnetwork from OD can be directly obtained post-training without the need to fine-tune, thus eliminating the requirement to access any labelled data.
+
+# 3.2 Training OD Formulation
+
+We propose two ways to train an OD-enabled network: i) plain $O D$ and ii) knowledge distillation $O D$ training (OD w/ KD). In the first approach, in each step we first sample $p \sim D _ { \mathcal { P } }$ ; then we perform the forward and backward pass using the $p$ -reduced network $\mathbf { F } _ { p }$ ; finally we update the submodel’s weights using the selected optimiser. Since sampling a $p$ -reduced network provides us significant computational savings on average, we can exploit this reduction to further boost accuracy. Therefore, in the second approach we exploit the nested structure of OD, i.e. $p _ { 1 } < p _ { 2 } \implies \mathbf { F } _ { p _ { 1 } } \subset \mathbf { F } _ { p _ { 2 } }$ and allow for the bigger capacity supermodel to teach the sampled $p$ -reduced network at each iteration via knowledge distillation (teacher $p _ { \operatorname* { m a x } } > p$ , $p _ { \operatorname* { m a x } } = \operatorname* { m a x } { \mathcal P }$ ). In particular, in each iteration, the loss function consists of two components as follows:
+
+![](images/89eb61d73c4481ca0ff08db1449e8b71e4f9769c77512ae027a9025ed7f69c13.jpg)  
+Figure 3: Full non-federated datasets. OD-Ordered Dropout with $D _ { \mathcal { P } } = \mathcal { U } _ { 5 }$ , SM-single independent models, KD-knowledge distillation.
+
+where $\mathrm { S M } _ { p }$ is the softmax output of the sampled $p$ -submodel, $\pmb { y } _ { \mathrm { l a b e l } }$ is the ground-truth label, CE is the cross-entropy function, KL is the KL divergence, $T$ is the distillation temperature [21] and $\alpha$ is the relative weight of the two components. We observed in our experiments always backpropagating also the teacher network further boosts performance. Furthermore, the best performing values for distillation were $\alpha = T = 1$ , thus smaller models exactly mimic the teacher output. This means that new knowledge propagates in submodels by proxy, i.e. by backpropagating on the teacher, leading to the following loss function:
+
+$$
+\mathscr { L } _ { d } ( \mathbf { S } \mathbf { M } _ { p } , \mathbf { S } \mathbf { M } _ { p _ { \operatorname* { m a x } } } , \pmb { y } _ { \mathrm { l a b e l } } ) = \mathbf { K L } ( \mathbf { S } \mathbf { M } _ { p } , \mathbf { S } \mathbf { M } _ { p _ { \operatorname* { m a x } } } , T ) + \mathbf { C } \mathbf { E } ( \operatorname* { m a x } ( \mathbf { S } \mathbf { M } _ { p _ { \operatorname* { m a x } } } ) , \pmb { y } _ { \mathrm { l a b e l } } )
+$$
+
+# 3.3 Ordered Dropout exactly recovers SVD
+
+We further show that our new OD formulation can recover the Singular Value Decomposition (SVD) in the case where there exists a linear mapping from features to responses. We formalise this claim in the following theorem.
+
+Theorem 1. Let $\mathbf { F } : \mathbb { R } ^ { n }  \mathbb { R } ^ { m }$ be a NN with two fully-connected linear layers with no activation or biases and $K = \operatorname* { m i n } \{ m , n \}$ hidden neurons. Moreover, let data $\mathcal { X }$ come from a uniform distribution on the $n$ -dimensional unit ball and $A$ be an $m \times n$ full rank matrix with $K$ distinct singular values. If response $y$ is linked to data $\mathcal { X }$ via a linear map: $x  A x$ and distribution $D _ { \mathcal { P } }$ is such that for every $b \in [ K ]$ there exists $p \in \mathcal P$ for which $b = \lceil p \cdot K \rceil$ , then for the optimal solution of
+
+$$
+\operatorname* { m i n } _ { \mathbf { F } } \mathbb { E } _ { { x } \sim { \mathcal { X } } } \mathbb { E } _ { { p } \sim { D _ { \mathcal { P } } } } \| \mathbf { F } _ { p } ( { x } ) - { y } \| ^ { 2 }
+$$
+
+it holds $\mathbf { F } _ { p } ( x ) = A _ { b } x$ , where $A _ { b }$ is the best $b$ -rank approximation of $A$ and $b = \lceil p \cdot K \rceil$ .
+
+Theorem 1 shows that our OD formulation exhibits not only intuitively, but also theoretically ordered importance representation. Proof of this claim is deferred to the Appendix.
+
+# 3.4 Model-Device Association
+
+Computational and Memory Implications. The primary objective of OD is to alleviate the excessive computational and memory demands of the training and inference deployments. When a layer is shrunk through OD, there is no need to perform the forward and backward passes or gradient updates on the pruned units. As a result, OD offers gains both in terms of FLOP count and model size. In particular, for every fully-connected and convolutional layer, the number of FLOPs and weight parameters is reduced by $K _ { 1 } { \cdot } K _ { 2 } \big / [ p { \cdot } K _ { 1 } ] { \cdot } [ p { \cdot } K _ { 2 } ] \sim 1 \big / p ^ { 2 }$ , where $K _ { 1 }$ and $K _ { 2 }$ correspond to the number of input and output neurons/channels, respectively. Accordingly, the bias terms are reduced by a factor of $K _ { 2 } / [ p \cdot K _ { 2 } ] \sim 1 / p$ . The normalisation, activation and pooling layers are compressed in terms of FLOPs and parameters similarly to the biases in fully-connected and convolutional layers. This is also evident in Table 1b. Finally, smaller model size also leads to reduced memory footprint for gradients and the optimiser’s state vectors such as momentum. However, how are these submodels related to devices in the wild and how is this getting modelled?
+
+Ordered Dropout Rates Space. Our primary objective with OD is to tackle device heterogeneity. Inherently, each device has certain capabilities and can run a specific number of model operations within a given time budget. Since each $p$ value defines a submodel of a given width, we can indirectly associate a $p _ { \mathrm { m a x } } ^ { i }$ value with the $i$ -th device capabilities, such as memory, processing throughput or energy budget. As such, each participating client is given at most the $p _ { \mathrm { m a x } } ^ { i ^ { \ast } }$ -submodel it can handle.
+
+Devices in the wild, however, can have dramatically different capabilities; a fact further exacerbated by the co-existence of previous-generation devices. Modelling discretely each device becomes quickly intractable at scale. Therefore, we cluster devices of similar capabilities together and subsequently associate a single $p _ { \mathrm { m a x } } ^ { i }$ value with each cluster. This clustering can be done heuristically (i.e. based on the specifications of the device) or via benchmarking of the model on the actual device and is considered a system-design decision for our paper. As smartphones nowadays run a multitude of simultaneous tasks [43], our framework can further support modelling of transient device load by reducing its associated $p _ { \mathrm { m a x } } ^ { i }$ , which essentially brings the capabilities of the device to a lower tier at run time, thus bringing real-time adaptability to FjORD.
+
+Concretely, the discrete candidate values of $\mathcal { P }$ depend on i) the number of clusters and corresponding device tiers, ii) the different load levels being modelled and iii) the size of the network itself, as i.e. for each tier $i$ there exists $p _ { \mathrm { m a x } } ^ { i }$ beyond which the network cannot be resolved. In this paper, we treat the former two as invariants (assumed to be given by the service provider), but provide results across different number and distributions of clusters, models and datasets.
+
+# 3.5 Preliminary Results
+
+Here, we present some results to showcase the performance of OD in the centralised non-FL training setting (i.e. the server has access to all training data) across three tasks, explained in detail in $\ S 5$ Concretely, we run OD with distribution $D _ { \mathcal { P } } = \mathcal { U } _ { 5 }$ (uniform distribution over the set $\{ i / 5 \} _ { i = 1 } ^ { 5 } ,$ ) and compare it with end-to-end trained submodels (SM) trained in isolation for the given width of the model. Fig. 3 shows that across the three datasets, the best attained performance of OD along every width $p$ is very close to the performance of the baseline models. We extend this comparison against Random Dropout in the Appendix. We note at this point that the submodel baselines are trained from scratch, explicitly optimised to that given width with no possibility to jump across them, while our OD model was trained using a single training loop and offers the ability to switch between accuracy-computation points without the need to retrain.
+
+#
+
+Building upon the shoulders of OD, we introduce FjORD, a framework for federated training over heterogenous clients. We subsequently describe the FjORD’s workflow, further documented in Alg. 1.
+
+As a starting point, the global model architecture, $\mathbf { F }$ , is initialised with weights $\pmb { w } ^ { 0 }$ , either randomly or via a pretrained network. The dropout rates space $\mathcal { P }$ is selected along with distribution $D _ { \mathcal { P } }$ with $| \mathcal { P } |$ discrete candidate values, with each $p$ corresponding to a subnetwork of the global model with varying FLOPs and parameters. Next, the devices to participate are clustered into $| \mathcal { C } _ { \mathrm { t i e r s } } |$ tiers and a $p _ { \mathrm { m a x } } ^ { c }$ value is associated with each cluster $c$ . The resulting $p _ { \mathrm { m a x } } ^ { c }$ represents the maximum capacity of the network that devices in this cluster can handle without violating a latency or memory constraint.
+
+At the beginning of each communication round $t$ , the set of participating devices $S _ { t }$ is determined, which either consists of all available clients $\boldsymbol { A } _ { t }$ or contains only a random subset of $\boldsymbol { A } _ { t }$ based on the server’s capacity. Next, the server broadcasts the current model to the set of clients $S _ { t }$ and each client $i$ receives ${ w _ { p } } _ { \operatorname* { m a x } } ^ { i }$ . On the client side, each client runs $E$ local iterations and at each local iteration $k$ , the device $i$ samples $p _ { ( i , k ) }$ from conditional distribution $D _ { \mathcal { P } } | D _ { \mathcal { P } } \leq p _ { \operatorname* { m a x } } ^ { i }$ which accounts for its limited capability. Subsequently, each client updates the respective weights $( { w _ { p } } _ { \boldsymbol { \mathrm { ( } i } , \boldsymbol { k } \mathrm { ) } } )$ of the local submodel using the FedAvg [46] update rule. In this step, other strategies [39, 63, 31] can be interchangeably employed. At the end of the local iterations, each device sends its update back to the server.
+
+Finally, the server aggregates these communicated changes and updates the global model, to be distributed in the next global federated round to a different subset of devices. Heterogeneity of devices leads to heterogeneity in the model updates and, hence, we need to account for that in the global aggregation step. To this end, we utilise the following aggregation rule
+
+$$
+\pmb { w } _ { s _ { j } } ^ { t + 1 } \backslash \pmb { w } _ { s _ { j - 1 } } ^ { t + 1 } = \mathrm { W A } \left( \left\{ \pmb { w } _ { i _ { s _ { j } } } ^ { ( i , t , E ) } \backslash \pmb { w } _ { s _ { j - 1 } } ^ { ( i , t , E ) } \right\} _ { i \in S _ { t } ^ { j } } \right)
+$$
+
+where ${ w _ { s } } _ { j . } \backslash w _ { s _ { j - 1 } }$ are the weights that belong to ${ \bf F } _ { s _ { j } }$ but not to $\mathbf { F } _ { { s } _ { j - 1 } }$ , ${ \pmb w } ^ { t + 1 }$ the global weights sj  sj−1at communication round $t + 1$ , $\mathbf { \Sigma } _ { w } ^ {  } ( i , t , E )$ sj  the weights on client $i$ sj−1 at communication round $t$ after $E$ local iterations, $S _ { t } ^ { j } = \{ i \in { \mathcal { S } } _ { t } : p _ { \operatorname* { m a x } } ^ { i } \geq s _ { j } \}$ a set of clients that have the capacity to update ${ \pmb w } _ { s _ { j } }$ , and WA stands for weighted average, where weights are proportional to the amount of data on each client.
+
+Communication Savings. In addition to the computational savings (§3.4), OD provides additional communication savings. First, for the server-to-client transfer, every device with $p _ { \mathrm { m a x } } ^ { i } < 1$ observes a reduction of approximately $^ 1 / ( p _ { \mathrm { m a x } } ^ { i } ) ^ { 2 }$ in the downstream transferred data due to the smaller model size $( \ S \ 3 . 4 )$ . Accordingly, the upstream client-to-server transfer is decreased by $^ 1 / ( p _ { \mathrm { m a x } } ^ { i } ) ^ { 2 }$ as only the gradient updates of the unpruned units are transmitted.
+
+# Algorithm 1: FjORD (Proposed Framework)
+
+Input: ${ \bf F } , { \pmb w } ^ { 0 } , D _ { \mathcal { P } } , T , E$   
+1 for $t \gets 0$ to $T - 1$ do // Global rounds   
+2 Server selects clients as a subset $S _ { t } \subset A _ { t }$   
+3 Server broadcasts weights of $p _ { \mathrm { m a x } } ^ { i }$ -submodel to each client $i \in S _ { t }$   
+4 for $k  0$ to $E - 1$ do // Local iterations   
+5 $\forall i \in S _ { t }$ : Device $_ { i }$ samples $p _ { ( i , k ) } \sim D p \vert D _ { \mathcal { P } } \leq p _ { \mathrm { m a x } } ^ { i }$ and updates the weights of local model   
+6 end   
+7 $\forall i \in S _ { t }$ : device $_ { i }$ sends to the server the updated weights $\mathbf { \Delta } _ { w } ^ { ( i , t , E ) }$   
+8 Server updates $\scriptstyle w ^ { t + 1 }$ as in Eq. (1)
+
+Identifiability. A standard procedure in FL is to perform element-wise averaging to aggregate model updates from clients. However, coordinate-wise averaging of updates may have detrimental effects on the accuracy of the global model, due to the permutation invariance of the hidden layers. Recent techniques tackle this problem by matching clients’ neurons before averaging [68, 57, 62]. Unfortunately, doing so is computationally expensive and hurts scalability. FjORD mitigates this issue since it exhibits the natural importance of neurons/channels within each hidden layer by design; essentially OD acts in lieu of a neuron matching algorithm without the computational overhead.
+
+Subnetwork Knowledge Transfer. In $\ S \ 3 . 2$ , we introduced knowledge distillation for our OD formulation. We extend this approach to FjORD, where instead of the full network, we employ width $\operatorname* { m a x } \{ p \in \mathcal { P } : p \leq p _ { \operatorname* { m a x } } ^ { i } \}$ as a teacher network in each local iteration on device $i$ . We provide the alternative of FjORD with knowledge distillation mainly as a solution for cases where the client bottleneck is memory- or network-related, rather than computational in nature [32]. However, in cases where client devices are computationally bound in terms of training latency, we propose FjORD without KD or decreasing $p _ { \mathrm { m a x } } ^ { i }$ to account for the overhead of KD.
+
+9 end   
+
+<table><tr><td>Dataset</td><td>Model</td><td># Clients # Samples Task</td></tr><tr><td>CIFAR10</td><td>ResNet18</td><td>10050,ooo Image classification</td></tr><tr><td>FEMNIST</td><td>CNN</td><td>3,400 671,585 Image classification</td></tr><tr><td>Shakespeare RNN</td><td></td><td>71538, OO1 Next character prediction</td></tr></table>
+
+# 5 Evaluation of FjORD
+
+In this section, we provide a thorough evaluation of FjORD and its components across different tasks, datasets, models and device cluster distributions to show its performance, elasticity and generality.
+
+Datasets and Models. We evaluate FjORD on two vision and one text prediction task, shown in Table 1a. For CIFAR10 [33], we use the “CIFAR” version of ResNet18 [20]. We federate the dataset by randomly dividing it into equally-sized partitions, each allocated to a specific client, and thus remaining IID in nature. For FEMNIST, we use a CNN with two convolutional layers followed by a softmax layer. For Shakespeare, we employ a RNN with an embedding layer (without dropout) followed by two LSTM [22] layers and a softmax layer. We report the model’s performance of the last epoch on the test set which is constructed by combining the test data for each client. We report top-1 accuracy vision tasks and negative perplexity for text prediction. Further details, such as hyperparameters, description of datasets and models are available in the Appendix.
+
+Table 1: Datasets and models   
+
+<table><tr><td colspan="2">p=0.2</td><td>0.4</td><td>0.6</td><td>0.8</td><td>1.0</td></tr><tr><td colspan="6">CIFAR10/ResNet18</td></tr><tr><td>MACs</td><td>23M</td><td>91M</td><td>203M</td><td>360M</td><td>555M</td></tr><tr><td>Params</td><td>456K</td><td>2M</td><td>4M</td><td>7M</td><td>11M</td></tr><tr><td colspan="6">FEMNIST/CNN</td></tr><tr><td>MACs</td><td>47K</td><td>120K</td><td>218K</td><td>342K</td><td>491K</td></tr><tr><td>Params</td><td>5K</td><td>10K</td><td>15K</td><td>20K</td><td>26K</td></tr><tr><td colspan="6">Shakespeare/RNN</td></tr><tr><td>MACs</td><td>12K</td><td>40K</td><td>83K</td><td>143K</td><td>216K</td></tr><tr><td>Params</td><td>12K</td><td>40K</td><td>82K</td><td>142K</td><td>214K</td></tr></table>
+
+# 5.1 Experimental Setup
+
+(b) MACs and parameters per $p$ -reduced network
+
+Infrastructure. FjORD was implemented on top of the Flower (v0.14dev) [5] framework and PyTorch (v1.4.0) [51]. We run all our experiments on a private cloud cluster, consisting of Nvidia V100 GPUs. To scale to hundreds of clients on a single machine, we optimized Flower so that clients only allocate GPU resources when actively participating in a federated client round. We report average performance and the standard deviation across three runs for all experiments. To model client availability, we run up to 100 Flower clients in parallel and sample $10 \%$ at each global round, with the ability for clients to switch identity at the beginning of each round to overprovision for larger federated datasets. Furthermore, we model client heterogeneity by assigning each client to one of the device clusters. We provide the following setups:
+
+![](images/71d5837d4dc2562a1840258a05df4e1d53d37bcf5b7b8222ff9e74cb7c6826d0.jpg)  
+Figure 4: Ordered Dropout with KD vs eFD baselines. Performance vs dropout rate $p$ across different networks and datasets. $D _ { \mathcal { P } } = \mathcal { U } _ { 5 }$
+
+Uniform-{5,10}: This refers to the distribution $D _ { \mathcal { P } }$ , i.e. $p \sim \mathcal { U } _ { k }$ , with $k = 5$ or 10.
+
+Drop Scale $\in \ \{ 0 . 5 , 1 . 0 \}$ : This parameter affects a possible skew in the number of devices per uniform- cluster. It refers to the drop in clients per cluster of devices, as we go to higher $n$ and drop scale $d s$ , the high-end cluster $n$ contains $\scriptstyle 1 - \sum _ { i = 0 } ^ { n - 1 } d s / _ { n }$ of the devices and the $p$ ’s. Formally, for rest of the clusters contain $\left. d s \right/ n$ each. Hence, for $d s { = } 1 . 0$ of the uniform-5 case, all devices can run the $p = 0 . 2$ subnetwork, $80 \%$ can run the $p = 0 . 4$ and so on, leading to a device distribution of $( 0 . 2 , . . . , 0 . 2 )$ . This percentage drop is half for the case of $d s { = } 0 . 5$ , resulting in a larger high-end cluster, e.g. $( 0 . 1 , 0 . 1 , . . . , 0 . 6 )$ .
+
+Baselines. To assess the performance against the state-of-the-art, we compare FjORD with the following baselines: i) Extended Federated Dropout (eFD), ii) FjORD with eFD (FjORD w/ eFD).
+
+eFD builds on top of the technique of Federated Dropout (FD) [8], which adopts a Random Dropout (RD) at neuron/filter level for minimising the model’s footprint. However, FD does not support adaptability to heterogeneous client capabilities out of the box, as it inherits a single dropout rate across devices. For this reason, we propose an extension to FD, allowing to adapt the dropout rate to the device capabilities, defined by the respective cluster membership. It is clear that eFD dominates FD in performance and provides a tougher baseline, as the latter needs to impose the same dropout rate to fit the model at hand on all devices, leading to larger dropout rates (i.e. uniform dropout of $80 \%$ for full model to support the low-end devices). We provide empirical evidence for this in the Appendix. For investigative purposes, we also applied eFD on top of FjORD, as a means to update a larger part of the model from lower-tier devices, i.e. allow them to evaluate submodels beyond their $p _ { \mathrm { m a x } } ^ { i }$ during training.
+
+# 5.2 Performance Evaluation
+
+In order to evaluate the performance of FjORD, we compare it to the two baselines, eFD and $_ \mathrm { O D + e F D }$ . We consider the uniform-5 setup with drop scale of 1.0 (i.e. uniform clusters). For each baseline, we train one independent model $\mathbf { F } _ { p }$ , end-to-end, for each $p$ . For eFD, what this translates to is that the clusters of devices that cannot run model $\mathbf { F } _ { p }$ compensate by randomly dropping out neurons/filters. We point out that $p = 0 . 2$ is omitted from the eFD results as it is essentially not employing any dropout whatsoever. For the case of $\mathrm { F j O R D + e F D }$ , we control the RD by capping it to $d = 0 . 2 5$ . This allows for larger submodels to be updated more often – as device belonging to cluster $c$ can now have pcmax $p _ { \mathrm { m a x } } ^ { c } \to p _ { \mathrm { m a x } } ^ { c + 1 }$ during training where $c { + 1 }$ is the next more powerful cluster – while at the same time it prevents the destructive effect of too high dropout values shown in the eFD baseline.
+
+Fig. 4 presents the achieved accuracy for varying values of $p$ across the three target datasets. FjORD (denoted by FjORD w/ KD) outperforms eFD across all datasets with improvements between 1.53- 34.87 percentage points (pp) (19.22 pp avg. across $p$ values) on CIFAR10, 1.57-6.27 pp (3.41 pp avg.) on FEMNIST and 0.01-0.82 points (p) (0.46 p avg.) on Shakespeare. Compared to FjORD $+ \mathrm { e F D }$ , FjORD achieves performance gains of $0 . 7 1 { \cdot } 2 . 6 6 \mathrm { p p }$ (1.79 avg.), up to 2.56 pp (1.35 pp avg.) on FEMNIST and $0 . 1 2 – 0 . 2 2 \mathfrak { p }$ $_ { 0 . 1 8 \mathrm { ~ p ~ } }$ avg.) on Shakespeare.
+
+Across all tasks, we observe that FjORD is able to improve its performance with increasing $p$ due to the nested structure of its OD method. We also conclude that eFD on top of FjORD does not seem to lead to better results. More importantly though, given the heterogeneous pool of devices, to obtain the highest performing model for eFD, multiple models have to be trained (i.e. one per device cluster). For instance, the highest performing models for eFD are $\mathbf { F } _ { 0 . 4 }$ , $\mathbf { F } _ { 0 . 6 }$ and ${ \bf F } _ { 0 . 4 }$ for CIFAR10, FEMNIST and Shakespeare respectively, which can be obtained only a posteriori; after all model variants have been trained. Instead, despite the device heterogeneity, FjORD requires a single training process that leads to a global model that significantly outperforms the best model of eFD (by 2.98 and $2 . 7 3 \mathrm { p p }$ for CIFAR10 and FEMNIST, respectively, and $0 . 1 3 { \mathfrak { p } }$ for Shakespeare), while allowing the direct, seamless extraction of submodels due to the nested structure of OD. Empirical evidence of the convergence of FjORD and the corresponding baselines is provided in the Appendix.
+
+![](images/39ce337186f7a1e3fdbf78655ebd1ab5be1e5d5d0469d21595f13b76dea224cb.jpg)  
+Figure 5: Ablation analysis of FjORD with Knowledge Distillation. Ordered Dropout with $D _ { \mathcal { P } } = \mathcal { U } _ { 5 }$ , KD - Knowledge distillation.
+
+# 5.3 Ablation Study of KD in FjORD
+
+To evaluate the contribution of our knowledge distillation method to the attainable performance of FjORD, we conduct an ablative analysis on all three datasets. We adopt the same setup of uniform-5 and drop scale $= 1 . 0$ as in the previous section and compare FjORD with and without KD.
+
+Fig. 5 shows the efficacy of FjORD’s KD in $\mathrm { F L }$ settings. FjORD’s KD consistently improves the performance across all three datasets when $p > 0 . 4$ , with average gains of 0.18, 0.68 and 0.87 pp for submodels of size 0.6, 0.8 and 1 on CIFAR-10, 1.96, 2.39 and 2.65 pp for FEMNIST and $0 . 1 0 \mathrm { p }$ for Shakespeare. For the cases of $p \leq 0 . 4$ , the impact of KD is fading. We believe this to be a side-effect of optimising for the average accuracy across submodels, which also yielded the $T = \alpha = 1$ strategy. We leave the exploration of alternative weighted KD strategies as future work. Overall, the use of KD significantly improves the performance of the global model, yielding gains of 0.71 and $2 . 6 3 \mathrm { p p }$ for CIFAR10 and FEMNIST and $0 . 1 0 \mathfrak { p }$ for Shakespeare.
+
+# 5.4 FjORD’s Deployment Flexibility
+
+# 5.4.1 Device Clusters Scalability
+
+An important characteristic of FjORD is its ability to scale to a larger number of device clusters or, equivalently, perform well with higher granularity of $p$ values. To illustrate this, we test the performance of OD across two setups, uniform-5 and $- I O$ (defined in $\ S 5 . 1$ ).
+
+As shown in Fig. 6, FjORD sustains its performance even under the higher granularity of $p$ values. This means that for applications where the modelling of clients needs to be more fine-grained, FjORD can still be of great value, without any significant degradation in achieved accuracy per submodel. This further supports the use-case where device-load needs to be modelled explicitly in device clusters (e.g. modelling device capabilities and load with deciles).
+
+# 5.4.2 Adaptability to Device Distributions
+
+In this section, we make a similar case about FjORD’s elasticity with respect to the allocation of available devices to each cluster. We adopt the setup of uniform-5 once again, but compare across drop scales 0.5 and 1.0 (defined in $\ S 5 . 1$ ). In both cases, clients that can support models of $p _ { \operatorname* { m a x } } ^ { i } \in \{ 0 . 2 , \dots , 0 . 8 \}$ are equisized, but the former halves the percentage of devices and allocates it to the last (high-end) cluster, now accounting for $60 \%$ of the devices. The rationale behind this is that the majority of participating devices are able to run the whole original model.
+
+The results depicted in Fig. 7 show that the larger submodels are expectedly more accurate, being updated more often. However, the same graphs also indicate that FjORD does not significantly degrade the accuracy of the smaller submodels in the presence of more high-tier devices (i. $e . \ d s = 0 . 5$ ). This is a direct consequence of sampling $p$ values during local rounds, instead of tying each tier with only the maximal submodel it can handle. We should also note that we did not alter the uniform sampling in this case on the premise that high-end devices are seen more often, precisely to illustrate FjORD’s adaptability to latent user device distribution changes of which the server may not be aware.
+
+# 6 Related Work
+
+Dropout Techniques. Contrary to conventional Random Dropout [59], which stochastically drops a different, random set of a layer’s units in every batch and is typically applied for regularisation purposes, OD employs a structured ordered dropping scheme that aims primarily at tunably reducing the computational and memory cost of training and inference. However, OD can still have an implicit regularisation effect since we encourage learning towards the top-ranked units (e.g. the left-most units in the example of Fig. 2), as these units will be dropped less often during training. Respectively, at inference time, the load of a client can be dynamically adjusted by dropping the least important units, i.e. adjusting the width of the network.
+
+![](images/0ca029dc94449dd6e2efe5b02648f369ca9bc267fb10e2bd80acb94520dbb0df.jpg)  
+Figure 6: Demonstration of FjORD’s scalability with respect to the number of device clusters.
+
+![](images/c9c92134ee38481d7e91a359f09b44762be14bcf1363ed4a2add3195c45ef0b3.jpg)  
+(b) RNN - Shakespeare   
+Figure 7: Demonstration of the adaptability of FjORD across different device distributions.
+
+To the best of our knowledge, the only similar technique to OD is Nested Dropout, where the authors proposed a similar construction, which is applied to the representation layer in autoencoders [54] in order to enforce identifiability of the learned representation or the last layer of the feature extractor [24] to learn an ordered set of features for transfer learning. In our case, we apply OD to every layer to elastically adapt the computation and memory requirements during training and inference.
+
+Traditional Pruning. Conventional non-FL compression techniques can be applicable to reduce the network size and computation needs. The majority of pruning methods [18, 16, 37, 35, 49] aim to generate a single pruned model and require access to labelled data in order to perform a costly fine-tuning/calibration for each pruned variant. Instead, FjORD’s Ordered Dropout enables the deterministic extraction of multiple pruned models with varying resource budgets directly after training. In this manner, we remove both the excessive overhead of fine-tuning and the need for labelled data availability, which is crucial for real-world, privacy-aware applications [60, 56]. Finally, other model compression methods [13, 64, 9] remain orthogonal to FjORD.
+
+System Heterogeneity. So far, although substantial effort has been devoted to alleviating the statistical heterogeneity [38] among clients [58, 36, 26, 12, 40], the system heterogeneity has largely remained unaddressed. Considering the diversity of client devices, techniques on client selection [50] and control of the per-round number of participating clients and local iterations [45, 65] have been developed. Nevertheless, as these schemes are restricted to allocate a uniform amount of work to each selected client, they either limit the model complexity to fit the lowest-end devices or exclude slow clients altogether. From an aggregation viewpoint, [39] allows for partial results to be integrated to the global model, thus enabling the allocation of different amounts of work across heterogeneous clients. Despite the fact that each client is allowed to perform a different number of local iterations based on its resources, large models still cannot be accommodated on the more constrained devices.
+
+Communication Optimisation. The majority of existing work has focused on tackling the communication overhead in FL. [32] proposed using structured and sketched updates to reduce the transmitted data. ATOMO [61] introduced a generalised gradient decomposition and sparsification technique, aiming to reduce the gradient sizes communicated upstream. [17] adaptively select the gradients’ sparsification degree based on the available bandwidth and computational power. Building upon gradient quantisation methods [44, 23, 53, 25], [2] proposed using quantisation in the model sharing and aggregation steps. However, their scheme requires the same clients to participate across all rounds, and is, thus, unsuitable for realistic settings where clients’ availability cannot be guaranteed. Despite the bandwidth savings, these communication-optimising approaches do not offer computational gains nor do they address device heterogeneity. Nonetheless, they remain orthogonal to our work and can be complementarily combined to further alleviate the communication cost.
+
+Computation-Communication Co-optimisation. A few works aim to co-optimise both the computational and bandwidth costs. PruneFL [29] proposes an unstructured pruning method. Despite the similarity to our work in terms of pruning, this method assumes a common pruned model across all clients at a given round, thus not allowing more powerful devices to update more weights. Hence, the pruned model needs to meet the constraints of the least capable devices, which severely limits the model capacity. Moreover, the adopted unstructured sparsity is difficult to translate to processing speed gains [67]. Federated Dropout [8] randomly sparsifies the global model, before sharing it to the clients. Similarly to PruneFL, Federated Dropout does not consider the system diversity and distributes the same model size to all clients. Thus, it is restricted by the low-end devices or excludes them altogether from the FL process. Additionally, Federated Dropout does not translate to computational benefits at inference time, since the whole model is deployed after federated training.
+
+Contrary to the presented works, our framework embraces the client heterogeneity, instead of treating it as a limitation, and thus pushes the boundaries of FL deployment in terms of fairness, scalability and performance by tailoring the model size to the device at hand, both at training and inference time, in a “train-once-deploy-everywhere” manner.
+
+# 7 Conclusions & Future Work
+
+In this work, we have introduced FjORD, a federated learning method for heterogeneous device training. To this direction, FjORD builds on top of our Ordered Dropout technique as a means to extract submodels of smaller footprints from a main model in a way where training the part also participates in training the whole. We show that our Ordered Dropout is equivalent to SVD for linear mappings and demonstrate that FjORD’s performance in the local and federated setting exceeds that of competing techniques, while maintaining flexibility across different environment setups.
+
+In the future, we plan to investigate how FjORD can be deployed and extended to future-gen devices and models in a life-long manner, the interplay between system and data heterogeneity for OD-based personalisation as well as alternative dynamic inference techniques for tackling system heterogeneity.
+
+# Broader Impact
+
+Our work has a dual broader societal impact: i) on privacy and fairness in participation and ii) on the environment. On the one hand, centralised DNN training [19] has been the norm for a long time, mainly facilitated by the advances in server-grade accelerator design and cheap storage. However, this paradigm comes with a set of disadvantages, both in terms of data privacy and energy consumption. With mobile and embedded devices becoming more capable and FL becoming a viable alternative [3, 7], one can leverage the free compute cycles of client devices to train models on-device, without data ever leaving the device premises. These devices, being typically battery-powered, operate under a more constrained power envelope compared to data-center accelerators [1]. Moreover, these devices are already deployed in the wild, but typically not used for training purposes. What FjORD contributes is the ability for even less capable devices to participate in the training process, thus increasing the representation of low-tier devices (and by extension the correlated demographic groups), as well as adding to the overall compute capabilities of the distributed system as a whole, potentially offsetting part of the carbon footprint of centralised training data centers [52].
+
+However, moving the computation cost from the service provider to the user of the device is a non-negligible step and the user should be made aware what their device is used for, especially if they are contributing to the knowledge of a model they do not own. Moreover, while many large data centers [11, 15] are increasingly dependent on renewable resources for meeting their power demands, this might not be the case for household electricity, which may impede the sustainability of training on device, at least in the short run.
+
+# Funding Disclosure
+
+This work was entirely performed at and funded by Samsung AI.
+
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parse/train/4fLr7H5D_eT/4fLr7H5D_eT_content_list.json

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+    {
+        "type": "text",
+        "text": "FjORD: Fair and Accurate Federated Learning under heterogeneous targets with Ordered Dropout ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Samuel Horváth∗ KAUST† Thuwal, KSA samuel.horvath@kaust.edu.sa ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Stefanos Laskaridis∗ Samsung AI Center Cambridge, UK mail@stefanos.cc ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mario Almeida∗   \nSamsung AI Center   \nCambridge, UK   \nmario.a@samsung.com ",
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+    {
+        "type": "text",
+        "text": "Ilias Leontiadis Samsung AI Center Cambridge, UK i.leontiadis@samsung.com ",
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+    {
+        "type": "text",
+        "text": "Stylianos I. Venieris Samsung AI Center Cambridge, UK s.venieris@samsung.com ",
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+    {
+        "type": "text",
+        "text": "Nicholas D. Lane Samsung AI Center &   \nUniversity of Cambridge Cambridge, UK   \nnic.lane@samsung.com ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Federated Learning (FL) has been gaining significant traction across different ML tasks, ranging from vision to keyboard predictions. In large-scale deployments, client heterogeneity is a fact and constitutes a primary problem for fairness, training performance and accuracy. Although significant efforts have been made into tackling statistical data heterogeneity, the diversity in the processing capabilities and network bandwidth of clients, termed as system heterogeneity, has remained largely unexplored. Current solutions either disregard a large portion of available devices or set a uniform limit on the model’s capacity, restricted by the least capable participants. In this work, we introduce Ordered Dropout, a mechanism that achieves an ordered, nested representation of knowledge in deep neural networks (DNNs) and enables the extraction of lower footprint submodels without the need of retraining. We further show that for linear maps our Ordered Dropout is equivalent to SVD. We employ this technique, along with a self-distillation methodology, in the realm of FL in a framework called FjORD. FjORD alleviates the problem of client system heterogeneity by tailoring the model width to the client’s capabilities. Extensive evaluation on both CNNs and RNNs across diverse modalities shows that FjORD consistently leads to significant performance gains over state-of-the-art baselines, while maintaining its nested structure. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Over the past few years, advances in deep learning have revolutionised the way we interact with everyday devices. Much of this success relies on the availability of large-scale training infrastructures and the collection of vast amounts of training data. However, users and providers are becoming increasingly aware of the privacy implications of this ever-increasing data collection, leading to the creation of various privacy-preserving initiatives by service providers [3] and government regulators [10]. ",
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+    {
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+        "text": "Federated Learning (FL) [46] is a relatively new subfield of machine learning (ML) that allows the training of models without the data leaving the users’ devices; instead, FL allows users to collaboratively train a model by moving the computation to them. At each round, participating devices download the latest model and compute an updated model using their local data. These locally trained models are then sent from the participating devices back to a central server where updates are aggregated for next round’s global model. Until now, a lot of research effort has been invested with the sole goal of maximising the accuracy of the global model [46, 42, 39, 31, 63], while complementary mechanisms have been proposed to ensure privacy and robustness [6, 14, 47, 48, 27, 4]. ",
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+        "image_caption": [
+            "Figure 1: FjORD employs OD to tailor the amount of computation to the capabilities of each participating device. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "image_caption": [
+            "Figure 2: Ordered vs. Random Dropout. In this example, the left-most features are used by more devices during training, creating a natural ordering to the importance of these features. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "A key challenge of deploying FL in the wild is the vast heterogeneity of devices [38], ranging from low-end IoT to flagship mobile devices. Despite this fact, the widely accepted norm in FL is that the local models have to share the same architecture as the global model. Under this assumption, developers typically opt to either drop low-tier devices from training, hence introducing training bias due to unseen data [30], or limit the global model’s size to accommodate the slowest clients, leading to degraded accuracy due to the restricted model capacity [8]. In addition to these limitations, variability in sample sizes, computation load and data transmission speeds further contribute to a very unbalanced training environment. Finally, the resulting model might not be as efficient as models specifically tailored to the capabilities of each device tier to meet the minimum processingperformance requirements [34]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work, we introduce FjORD (Fig. 1), a novel adaptive training framework that enables heterogeneous devices to participate in FL by dynamically adapting model size – and thus computation, memory and data exchange sizes – to the available client resources. To this end, we introduce Ordered Dropout (OD), a mechanism for run-time ordered (importance-based) pruning, which enables us to extract and train submodels in a nested manner. As such, OD enables all devices to participate in the FL process independently of their capabilities by training a submodel of the original DNN, while still contributing knowledge to the global model. Alongside OD, we propose a self-distillation method from the maximal supported submodel on a device to enhance the feature extraction of smaller submodels. Finally, our framework has the additional benefit of producing models that can be dynamically scaled during inference, based on the hardware and load constraints of the device. ",
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+    {
+        "type": "text",
+        "text": "Our evaluation shows that FjORD enables significant accuracy benefits over the baselines across diverse datasets and networks, while allowing for the extraction of submodels of varying FLOPs and sizes without the need for retraining. ",
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+    {
+        "type": "text",
+        "text": "2 Motivation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Despite the progress on the accuracy front, the unique deployment challenges of FL still set a limit to the attainable performance. FL is typically deployed on either siloed setups, such as among hospitals, or on mobile devices in the wild [7]. In this work, we focus on the latter setting. Hence, while cloud-based distributed training uses powerful high-end clients [19], in FL these are commonly substituted by resource-constrained and heterogeneous embedded devices. ",
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+    {
+        "type": "text",
+        "text": "In this respect, FL deployment is currently hindered by the vast heterogeneity of client hardware [66, 28, 7]. On the one hand, different mobile hardware leads to significantly varying processing speed [1], in turn leading to longer waits upon aggregation of updates (i.e. stragglers). At the same time, devices of mid and low tiers might not even be able to support larger models, e.g. the model does not fit in memory or processing is slow, and, thus, are either excluded or dropped upon timeouts from the training process, together with their unique data. More interestingly, the resource allocation to participating devices may also reflect on demographic and socio-economic information of owners, that makes the exclusion of such clients unfair [30] in terms of participation. Analogous to the device load and heterogeneity, a similar trend can be traced in the downstream (model) and upstream (updates) network communication in FL, which can be an additional substantial bottleneck for the training procedure [55]. ",
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+        "type": "text",
+        "text": "3 Ordered Dropout ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "In this paper, we firstly introduce the tools that act as enablers for heterogeneous federated training. Concretely, we have devised a mechanism of importance-based pruning for the easy extraction of subnetworks from the original, specially trained model, each with a different computational and memory footprint. We name this technique Ordered Dropout (OD), as it orders knowledge representation in nested submodels of the original network. ",
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+        "text": "More specifically, our technique starts by sampling a value (denoted by $p$ ) from a distribution of candidate values. Each of these values corresponds to a specific submodel, which in turn gets translated to a specific computational and memory footprint (see Table 1b). Such sampled values and associations are depicted in Fig. 2. Contrary to conventional dropout (RD), our technique drops adjacent components of the model instead of random neurons, which translates to computational benefits3 in today’s linear algebra libraries and higher accuracy as shown later. ",
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+    {
+        "type": "text",
+        "text": "3.1 Ordered Dropout Mechanics ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The proposed OD method is parametrised with respect to: i) the value of the dropout rate $p \\in ( 0 , 1 ]$ per layer, ii) the set of candidate values $\\mathcal { P }$ , such that $p \\in \\mathcal P$ and iii) the sampling method of $p$ over the set of candidate values, such that $p \\sim D _ { \\mathcal { P } }$ , where $D _ { \\mathcal { P } }$ is the distribution over $\\mathcal { P }$ . ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "A primary hyperparameter of OD is the dropout rate $p$ which defines how much of each layer is to be included, with the rest of the units dropped in a structured and ordered manner. The value of $p$ is selected by sampling from the dropout distribution $D _ { \\mathcal { P } }$ which is represented by a set of discrete values $\\mathcal { P } = \\{ s _ { 1 } , s _ { 2 } , . . . , s _ { | \\mathcal { P } | } \\}$ such that $0 { < } s _ { 1 } { < } . . . { < } s _ { | } p _ { | } \\leq 1$ and probabilities $\\mathbf { P } ( { \\dot { p } } = s _ { i } ) > 0$ , $\\forall i \\in [ | \\mathcal { P } | ]$ such that $\\begin{array} { r } { \\sum _ { i = 1 } ^ { | \\mathcal { P } | } \\mathbf { P } ( p = s _ { i } ) = 1 } \\end{array}$ . For instance, a uniform distribution over $\\mathcal { P }$ is denoted by $p \\sim \\mathcal { U } _ { \\mathcal { P } }$ (i.e. $D = \\mathcal { U }$ ). In our experiments we use uniform distribution over the set $\\mathcal { P } = \\{ i /  { k } \\} _ { i = 1 } ^ { k }$ , which we refer to as $\\mathcal { U } _ { k }$ (or uniform- $k$ ). The discrete nature of the distribution stems from the innately discrete number of neurons or filters to be selected. The selection of set $\\mathcal { P }$ is discussed in the next subsection. ",
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+    },
+    {
+        "type": "text",
+        "text": "The dropout rate $p$ can be constant across all layers or configured individually per layer $l$ , leading to $p _ { l } \\sim \\dot { D } _ { \\mathcal { P } } ^ { l }$ . As such an approach opens the search space dramatically, we refer the reader to NAS techniques [69] and continue with the same $p$ value across network layers for simplicity, without hurting the generality of our approach. ",
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+    {
+        "type": "text",
+        "text": "Given a $p$ value, a pruned $p$ -subnetwork can be directly obtained as follows. For each4 layer $l$ with width5 $K _ { l }$ , the submodel for a given $p$ has all neurons/filters with index $\\{ 0 , 1 , \\ldots , \\lceil p \\cdot K _ { l } \\rceil - 1 \\}$ included and $\\{ \\lceil p \\cdot K _ { l } \\rceil , \\ldots , K _ { l } ^ { - } - 1 \\}$ pruned. Moreover, the unnecessary connections between pruned neurons/filters are also removed6. We denote a pruned $p$ -subnetwork $\\mathbf { F } _ { p }$ with its weights ${ \\pmb w } _ { p }$ , where $\\mathbf { F }$ and $\\pmb { w }$ are the original network and weights, respectively. Importantly, contrary to existing pruning techniques [18, 35, 49], a $p$ -subnetwork from OD can be directly obtained post-training without the need to fine-tune, thus eliminating the requirement to access any labelled data. ",
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+    {
+        "type": "text",
+        "text": "3.2 Training OD Formulation ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We propose two ways to train an OD-enabled network: i) plain $O D$ and ii) knowledge distillation $O D$ training (OD w/ KD). In the first approach, in each step we first sample $p \\sim D _ { \\mathcal { P } }$ ; then we perform the forward and backward pass using the $p$ -reduced network $\\mathbf { F } _ { p }$ ; finally we update the submodel’s weights using the selected optimiser. Since sampling a $p$ -reduced network provides us significant computational savings on average, we can exploit this reduction to further boost accuracy. Therefore, in the second approach we exploit the nested structure of OD, i.e. $p _ { 1 } < p _ { 2 } \\implies \\mathbf { F } _ { p _ { 1 } } \\subset \\mathbf { F } _ { p _ { 2 } }$ and allow for the bigger capacity supermodel to teach the sampled $p$ -reduced network at each iteration via knowledge distillation (teacher $p _ { \\operatorname* { m a x } } > p$ , $p _ { \\operatorname* { m a x } } = \\operatorname* { m a x } { \\mathcal P }$ ). In particular, in each iteration, the loss function consists of two components as follows: ",
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+        "img_path": "images/89eb61d73c4481ca0ff08db1449e8b71e4f9769c77512ae027a9025ed7f69c13.jpg",
+        "image_caption": [
+            "Figure 3: Full non-federated datasets. OD-Ordered Dropout with $D _ { \\mathcal { P } } = \\mathcal { U } _ { 5 }$ , SM-single independent models, KD-knowledge distillation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathrm { S M } _ { p }$ is the softmax output of the sampled $p$ -submodel, $\\pmb { y } _ { \\mathrm { l a b e l } }$ is the ground-truth label, CE is the cross-entropy function, KL is the KL divergence, $T$ is the distillation temperature [21] and $\\alpha$ is the relative weight of the two components. We observed in our experiments always backpropagating also the teacher network further boosts performance. Furthermore, the best performing values for distillation were $\\alpha = T = 1$ , thus smaller models exactly mimic the teacher output. This means that new knowledge propagates in submodels by proxy, i.e. by backpropagating on the teacher, leading to the following loss function: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/800c65c338f4421cd2d1fb2e2145bab46d924088955cedb228d4ddb490803138.jpg",
+        "text": "$$\n\\mathscr { L } _ { d } ( \\mathbf { S } \\mathbf { M } _ { p } , \\mathbf { S } \\mathbf { M } _ { p _ { \\operatorname* { m a x } } } , \\pmb { y } _ { \\mathrm { l a b e l } } ) = \\mathbf { K L } ( \\mathbf { S } \\mathbf { M } _ { p } , \\mathbf { S } \\mathbf { M } _ { p _ { \\operatorname* { m a x } } } , T ) + \\mathbf { C } \\mathbf { E } ( \\operatorname* { m a x } ( \\mathbf { S } \\mathbf { M } _ { p _ { \\operatorname* { m a x } } } ) , \\pmb { y } _ { \\mathrm { l a b e l } } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "3.3 Ordered Dropout exactly recovers SVD ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We further show that our new OD formulation can recover the Singular Value Decomposition (SVD) in the case where there exists a linear mapping from features to responses. We formalise this claim in the following theorem. ",
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Let $\\mathbf { F } : \\mathbb { R } ^ { n }  \\mathbb { R } ^ { m }$ be a NN with two fully-connected linear layers with no activation or biases and $K = \\operatorname* { m i n } \\{ m , n \\}$ hidden neurons. Moreover, let data $\\mathcal { X }$ come from a uniform distribution on the $n$ -dimensional unit ball and $A$ be an $m \\times n$ full rank matrix with $K$ distinct singular values. If response $y$ is linked to data $\\mathcal { X }$ via a linear map: $x  A x$ and distribution $D _ { \\mathcal { P } }$ is such that for every $b \\in [ K ]$ there exists $p \\in \\mathcal P$ for which $b = \\lceil p \\cdot K \\rceil$ , then for the optimal solution of ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a735839809d01499da3dcbac585499bdf0d40785599ecf5bf0d936e94beaa50c.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { F } } \\mathbb { E } _ { { x } \\sim { \\mathcal { X } } } \\mathbb { E } _ { { p } \\sim { D _ { \\mathcal { P } } } } \\| \\mathbf { F } _ { p } ( { x } ) - { y } \\| ^ { 2 }\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "it holds $\\mathbf { F } _ { p } ( x ) = A _ { b } x$ , where $A _ { b }$ is the best $b$ -rank approximation of $A$ and $b = \\lceil p \\cdot K \\rceil$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 1 shows that our OD formulation exhibits not only intuitively, but also theoretically ordered importance representation. Proof of this claim is deferred to the Appendix. ",
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+    {
+        "type": "text",
+        "text": "3.4 Model-Device Association ",
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+        "type": "text",
+        "text": "Computational and Memory Implications. The primary objective of OD is to alleviate the excessive computational and memory demands of the training and inference deployments. When a layer is shrunk through OD, there is no need to perform the forward and backward passes or gradient updates on the pruned units. As a result, OD offers gains both in terms of FLOP count and model size. In particular, for every fully-connected and convolutional layer, the number of FLOPs and weight parameters is reduced by $K _ { 1 } { \\cdot } K _ { 2 } \\big / [ p { \\cdot } K _ { 1 } ] { \\cdot } [ p { \\cdot } K _ { 2 } ] \\sim 1 \\big / p ^ { 2 }$ , where $K _ { 1 }$ and $K _ { 2 }$ correspond to the number of input and output neurons/channels, respectively. Accordingly, the bias terms are reduced by a factor of $K _ { 2 } / [ p \\cdot K _ { 2 } ] \\sim 1 / p$ . The normalisation, activation and pooling layers are compressed in terms of FLOPs and parameters similarly to the biases in fully-connected and convolutional layers. This is also evident in Table 1b. Finally, smaller model size also leads to reduced memory footprint for gradients and the optimiser’s state vectors such as momentum. However, how are these submodels related to devices in the wild and how is this getting modelled? ",
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+        "type": "text",
+        "text": "Ordered Dropout Rates Space. Our primary objective with OD is to tackle device heterogeneity. Inherently, each device has certain capabilities and can run a specific number of model operations within a given time budget. Since each $p$ value defines a submodel of a given width, we can indirectly associate a $p _ { \\mathrm { m a x } } ^ { i }$ value with the $i$ -th device capabilities, such as memory, processing throughput or energy budget. As such, each participating client is given at most the $p _ { \\mathrm { m a x } } ^ { i ^ { \\ast } }$ -submodel it can handle. ",
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+        "text": "Devices in the wild, however, can have dramatically different capabilities; a fact further exacerbated by the co-existence of previous-generation devices. Modelling discretely each device becomes quickly intractable at scale. Therefore, we cluster devices of similar capabilities together and subsequently associate a single $p _ { \\mathrm { m a x } } ^ { i }$ value with each cluster. This clustering can be done heuristically (i.e. based on the specifications of the device) or via benchmarking of the model on the actual device and is considered a system-design decision for our paper. As smartphones nowadays run a multitude of simultaneous tasks [43], our framework can further support modelling of transient device load by reducing its associated $p _ { \\mathrm { m a x } } ^ { i }$ , which essentially brings the capabilities of the device to a lower tier at run time, thus bringing real-time adaptability to FjORD. ",
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+        "text": "Concretely, the discrete candidate values of $\\mathcal { P }$ depend on i) the number of clusters and corresponding device tiers, ii) the different load levels being modelled and iii) the size of the network itself, as i.e. for each tier $i$ there exists $p _ { \\mathrm { m a x } } ^ { i }$ beyond which the network cannot be resolved. In this paper, we treat the former two as invariants (assumed to be given by the service provider), but provide results across different number and distributions of clusters, models and datasets. ",
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+        "text": "3.5 Preliminary Results ",
+        "text_level": 1,
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+        "text": "Here, we present some results to showcase the performance of OD in the centralised non-FL training setting (i.e. the server has access to all training data) across three tasks, explained in detail in $\\ S 5$ Concretely, we run OD with distribution $D _ { \\mathcal { P } } = \\mathcal { U } _ { 5 }$ (uniform distribution over the set $\\{ i / 5 \\} _ { i = 1 } ^ { 5 } ,$ ) and compare it with end-to-end trained submodels (SM) trained in isolation for the given width of the model. Fig. 3 shows that across the three datasets, the best attained performance of OD along every width $p$ is very close to the performance of the baseline models. We extend this comparison against Random Dropout in the Appendix. We note at this point that the submodel baselines are trained from scratch, explicitly optimised to that given width with no possibility to jump across them, while our OD model was trained using a single training loop and offers the ability to switch between accuracy-computation points without the need to retrain. ",
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+        "text": "Building upon the shoulders of OD, we introduce FjORD, a framework for federated training over heterogenous clients. We subsequently describe the FjORD’s workflow, further documented in Alg. 1. ",
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+        "text": "As a starting point, the global model architecture, $\\mathbf { F }$ , is initialised with weights $\\pmb { w } ^ { 0 }$ , either randomly or via a pretrained network. The dropout rates space $\\mathcal { P }$ is selected along with distribution $D _ { \\mathcal { P } }$ with $| \\mathcal { P } |$ discrete candidate values, with each $p$ corresponding to a subnetwork of the global model with varying FLOPs and parameters. Next, the devices to participate are clustered into $| \\mathcal { C } _ { \\mathrm { t i e r s } } |$ tiers and a $p _ { \\mathrm { m a x } } ^ { c }$ value is associated with each cluster $c$ . The resulting $p _ { \\mathrm { m a x } } ^ { c }$ represents the maximum capacity of the network that devices in this cluster can handle without violating a latency or memory constraint. ",
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+        "text": "At the beginning of each communication round $t$ , the set of participating devices $S _ { t }$ is determined, which either consists of all available clients $\\boldsymbol { A } _ { t }$ or contains only a random subset of $\\boldsymbol { A } _ { t }$ based on the server’s capacity. Next, the server broadcasts the current model to the set of clients $S _ { t }$ and each client $i$ receives ${ w _ { p } } _ { \\operatorname* { m a x } } ^ { i }$ . On the client side, each client runs $E$ local iterations and at each local iteration $k$ , the device $i$ samples $p _ { ( i , k ) }$ from conditional distribution $D _ { \\mathcal { P } } | D _ { \\mathcal { P } } \\leq p _ { \\operatorname* { m a x } } ^ { i }$ which accounts for its limited capability. Subsequently, each client updates the respective weights $( { w _ { p } } _ { \\boldsymbol { \\mathrm { ( } i } , \\boldsymbol { k } \\mathrm { ) } } )$ of the local submodel using the FedAvg [46] update rule. In this step, other strategies [39, 63, 31] can be interchangeably employed. At the end of the local iterations, each device sends its update back to the server. ",
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+        "text": "Finally, the server aggregates these communicated changes and updates the global model, to be distributed in the next global federated round to a different subset of devices. Heterogeneity of devices leads to heterogeneity in the model updates and, hence, we need to account for that in the global aggregation step. To this end, we utilise the following aggregation rule ",
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+        "type": "equation",
+        "img_path": "images/2e5ef12564252cef4146e059f013850f4af17cf094ee5cd227c2ddf8ea598f76.jpg",
+        "text": "$$\n\\pmb { w } _ { s _ { j } } ^ { t + 1 } \\backslash \\pmb { w } _ { s _ { j - 1 } } ^ { t + 1 } = \\mathrm { W A } \\left( \\left\\{ \\pmb { w } _ { i _ { s _ { j } } } ^ { ( i , t , E ) } \\backslash \\pmb { w } _ { s _ { j - 1 } } ^ { ( i , t , E ) } \\right\\} _ { i \\in S _ { t } ^ { j } } \\right)\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where ${ w _ { s } } _ { j . } \\backslash w _ { s _ { j - 1 } }$ are the weights that belong to ${ \\bf F } _ { s _ { j } }$ but not to $\\mathbf { F } _ { { s } _ { j - 1 } }$ , ${ \\pmb w } ^ { t + 1 }$ the global weights sj  sj−1at communication round $t + 1$ , $\\mathbf { \\Sigma } _ { w } ^ {  } ( i , t , E )$ sj  the weights on client $i$ sj−1 at communication round $t$ after $E$ local iterations, $S _ { t } ^ { j } = \\{ i \\in { \\mathcal { S } } _ { t } : p _ { \\operatorname* { m a x } } ^ { i } \\geq s _ { j } \\}$ a set of clients that have the capacity to update ${ \\pmb w } _ { s _ { j } }$ , and WA stands for weighted average, where weights are proportional to the amount of data on each client. ",
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+        "text": "Communication Savings. In addition to the computational savings (§3.4), OD provides additional communication savings. First, for the server-to-client transfer, every device with $p _ { \\mathrm { m a x } } ^ { i } < 1$ observes a reduction of approximately $^ 1 / ( p _ { \\mathrm { m a x } } ^ { i } ) ^ { 2 }$ in the downstream transferred data due to the smaller model size $( \\ S \\ 3 . 4 )$ . Accordingly, the upstream client-to-server transfer is decreased by $^ 1 / ( p _ { \\mathrm { m a x } } ^ { i } ) ^ { 2 }$ as only the gradient updates of the unpruned units are transmitted. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1: FjORD (Proposed Framework) ",
+        "text_level": 1,
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+        "text": "Input: ${ \\bf F } , { \\pmb w } ^ { 0 } , D _ { \\mathcal { P } } , T , E$   \n1 for $t \\gets 0$ to $T - 1$ do // Global rounds   \n2 Server selects clients as a subset $S _ { t } \\subset A _ { t }$   \n3 Server broadcasts weights of $p _ { \\mathrm { m a x } } ^ { i }$ -submodel to each client $i \\in S _ { t }$   \n4 for $k  0$ to $E - 1$ do // Local iterations   \n5 $\\forall i \\in S _ { t }$ : Device $_ { i }$ samples $p _ { ( i , k ) } \\sim D p \\vert D _ { \\mathcal { P } } \\leq p _ { \\mathrm { m a x } } ^ { i }$ and updates the weights of local model   \n6 end   \n7 $\\forall i \\in S _ { t }$ : device $_ { i }$ sends to the server the updated weights $\\mathbf { \\Delta } _ { w } ^ { ( i , t , E ) }$   \n8 Server updates $\\scriptstyle w ^ { t + 1 }$ as in Eq. (1) ",
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+        "text": "Identifiability. A standard procedure in FL is to perform element-wise averaging to aggregate model updates from clients. However, coordinate-wise averaging of updates may have detrimental effects on the accuracy of the global model, due to the permutation invariance of the hidden layers. Recent techniques tackle this problem by matching clients’ neurons before averaging [68, 57, 62]. Unfortunately, doing so is computationally expensive and hurts scalability. FjORD mitigates this issue since it exhibits the natural importance of neurons/channels within each hidden layer by design; essentially OD acts in lieu of a neuron matching algorithm without the computational overhead. ",
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+        "type": "text",
+        "text": "Subnetwork Knowledge Transfer. In $\\ S \\ 3 . 2$ , we introduced knowledge distillation for our OD formulation. We extend this approach to FjORD, where instead of the full network, we employ width $\\operatorname* { m a x } \\{ p \\in \\mathcal { P } : p \\leq p _ { \\operatorname* { m a x } } ^ { i } \\}$ as a teacher network in each local iteration on device $i$ . We provide the alternative of FjORD with knowledge distillation mainly as a solution for cases where the client bottleneck is memory- or network-related, rather than computational in nature [32]. However, in cases where client devices are computationally bound in terms of training latency, we propose FjORD without KD or decreasing $p _ { \\mathrm { m a x } } ^ { i }$ to account for the overhead of KD. ",
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+    {
+        "type": "table",
+        "img_path": "images/0ec9f1b2172d903b23f91a38e433f1f58e3bbb058fbece350e9b293a663b38c6.jpg",
+        "table_caption": [
+            "9 end "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Model</td><td># Clients # Samples Task</td></tr><tr><td>CIFAR10</td><td>ResNet18</td><td>10050,ooo Image classification</td></tr><tr><td>FEMNIST</td><td>CNN</td><td>3,400 671,585 Image classification</td></tr><tr><td>Shakespeare RNN</td><td></td><td>71538, OO1 Next character prediction</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 Evaluation of FjORD ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we provide a thorough evaluation of FjORD and its components across different tasks, datasets, models and device cluster distributions to show its performance, elasticity and generality. ",
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+        "type": "text",
+        "text": "Datasets and Models. We evaluate FjORD on two vision and one text prediction task, shown in Table 1a. For CIFAR10 [33], we use the “CIFAR” version of ResNet18 [20]. We federate the dataset by randomly dividing it into equally-sized partitions, each allocated to a specific client, and thus remaining IID in nature. For FEMNIST, we use a CNN with two convolutional layers followed by a softmax layer. For Shakespeare, we employ a RNN with an embedding layer (without dropout) followed by two LSTM [22] layers and a softmax layer. We report the model’s performance of the last epoch on the test set which is constructed by combining the test data for each client. We report top-1 accuracy vision tasks and negative perplexity for text prediction. Further details, such as hyperparameters, description of datasets and models are available in the Appendix. ",
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+    {
+        "type": "table",
+        "img_path": "images/0c58f2fcaeb3a319d68c2fad79cdfc9cb2c77d60d0e8823a223655d5069032fa.jpg",
+        "table_caption": [
+            "Table 1: Datasets and models "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">p=0.2</td><td>0.4</td><td>0.6</td><td>0.8</td><td>1.0</td></tr><tr><td colspan=\"6\">CIFAR10/ResNet18</td></tr><tr><td>MACs</td><td>23M</td><td>91M</td><td>203M</td><td>360M</td><td>555M</td></tr><tr><td>Params</td><td>456K</td><td>2M</td><td>4M</td><td>7M</td><td>11M</td></tr><tr><td colspan=\"6\">FEMNIST/CNN</td></tr><tr><td>MACs</td><td>47K</td><td>120K</td><td>218K</td><td>342K</td><td>491K</td></tr><tr><td>Params</td><td>5K</td><td>10K</td><td>15K</td><td>20K</td><td>26K</td></tr><tr><td colspan=\"6\">Shakespeare/RNN</td></tr><tr><td>MACs</td><td>12K</td><td>40K</td><td>83K</td><td>143K</td><td>216K</td></tr><tr><td>Params</td><td>12K</td><td>40K</td><td>82K</td><td>142K</td><td>214K</td></tr></table>",
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+        "type": "text",
+        "text": "5.1 Experimental Setup ",
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+        "type": "text",
+        "text": "(b) MACs and parameters per $p$ -reduced network ",
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+        "type": "text",
+        "text": "Infrastructure. FjORD was implemented on top of the Flower (v0.14dev) [5] framework and PyTorch (v1.4.0) [51]. We run all our experiments on a private cloud cluster, consisting of Nvidia V100 GPUs. To scale to hundreds of clients on a single machine, we optimized Flower so that clients only allocate GPU resources when actively participating in a federated client round. We report average performance and the standard deviation across three runs for all experiments. To model client availability, we run up to 100 Flower clients in parallel and sample $10 \\%$ at each global round, with the ability for clients to switch identity at the beginning of each round to overprovision for larger federated datasets. Furthermore, we model client heterogeneity by assigning each client to one of the device clusters. We provide the following setups: ",
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+        "image_caption": [
+            "Figure 4: Ordered Dropout with KD vs eFD baselines. Performance vs dropout rate $p$ across different networks and datasets. $D _ { \\mathcal { P } } = \\mathcal { U } _ { 5 }$ "
+        ],
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+        "type": "text",
+        "text": "Uniform-{5,10}: This refers to the distribution $D _ { \\mathcal { P } }$ , i.e. $p \\sim \\mathcal { U } _ { k }$ , with $k = 5$ or 10. ",
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+        "text": "Drop Scale $\\in \\ \\{ 0 . 5 , 1 . 0 \\}$ : This parameter affects a possible skew in the number of devices per uniform- cluster. It refers to the drop in clients per cluster of devices, as we go to higher $n$ and drop scale $d s$ , the high-end cluster $n$ contains $\\scriptstyle 1 - \\sum _ { i = 0 } ^ { n - 1 } d s / _ { n }$ of the devices and the $p$ ’s. Formally, for rest of the clusters contain $\\left. d s \\right/ n$ each. Hence, for $d s { = } 1 . 0$ of the uniform-5 case, all devices can run the $p = 0 . 2$ subnetwork, $80 \\%$ can run the $p = 0 . 4$ and so on, leading to a device distribution of $( 0 . 2 , . . . , 0 . 2 )$ . This percentage drop is half for the case of $d s { = } 0 . 5$ , resulting in a larger high-end cluster, e.g. $( 0 . 1 , 0 . 1 , . . . , 0 . 6 )$ . ",
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+        "text": "Baselines. To assess the performance against the state-of-the-art, we compare FjORD with the following baselines: i) Extended Federated Dropout (eFD), ii) FjORD with eFD (FjORD w/ eFD). ",
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+        "text": "eFD builds on top of the technique of Federated Dropout (FD) [8], which adopts a Random Dropout (RD) at neuron/filter level for minimising the model’s footprint. However, FD does not support adaptability to heterogeneous client capabilities out of the box, as it inherits a single dropout rate across devices. For this reason, we propose an extension to FD, allowing to adapt the dropout rate to the device capabilities, defined by the respective cluster membership. It is clear that eFD dominates FD in performance and provides a tougher baseline, as the latter needs to impose the same dropout rate to fit the model at hand on all devices, leading to larger dropout rates (i.e. uniform dropout of $80 \\%$ for full model to support the low-end devices). We provide empirical evidence for this in the Appendix. For investigative purposes, we also applied eFD on top of FjORD, as a means to update a larger part of the model from lower-tier devices, i.e. allow them to evaluate submodels beyond their $p _ { \\mathrm { m a x } } ^ { i }$ during training. ",
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+        "type": "text",
+        "text": "5.2 Performance Evaluation ",
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+        "type": "text",
+        "text": "In order to evaluate the performance of FjORD, we compare it to the two baselines, eFD and $_ \\mathrm { O D + e F D }$ . We consider the uniform-5 setup with drop scale of 1.0 (i.e. uniform clusters). For each baseline, we train one independent model $\\mathbf { F } _ { p }$ , end-to-end, for each $p$ . For eFD, what this translates to is that the clusters of devices that cannot run model $\\mathbf { F } _ { p }$ compensate by randomly dropping out neurons/filters. We point out that $p = 0 . 2$ is omitted from the eFD results as it is essentially not employing any dropout whatsoever. For the case of $\\mathrm { F j O R D + e F D }$ , we control the RD by capping it to $d = 0 . 2 5$ . This allows for larger submodels to be updated more often – as device belonging to cluster $c$ can now have pcmax $p _ { \\mathrm { m a x } } ^ { c } \\to p _ { \\mathrm { m a x } } ^ { c + 1 }$ during training where $c { + 1 }$ is the next more powerful cluster – while at the same time it prevents the destructive effect of too high dropout values shown in the eFD baseline. ",
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+        "text": "Fig. 4 presents the achieved accuracy for varying values of $p$ across the three target datasets. FjORD (denoted by FjORD w/ KD) outperforms eFD across all datasets with improvements between 1.53- 34.87 percentage points (pp) (19.22 pp avg. across $p$ values) on CIFAR10, 1.57-6.27 pp (3.41 pp avg.) on FEMNIST and 0.01-0.82 points (p) (0.46 p avg.) on Shakespeare. Compared to FjORD $+ \\mathrm { e F D }$ , FjORD achieves performance gains of $0 . 7 1 { \\cdot } 2 . 6 6 \\mathrm { p p }$ (1.79 avg.), up to 2.56 pp (1.35 pp avg.) on FEMNIST and $0 . 1 2 – 0 . 2 2 \\mathfrak { p }$ $_ { 0 . 1 8 \\mathrm { ~ p ~ } }$ avg.) on Shakespeare. ",
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+        "text": "Across all tasks, we observe that FjORD is able to improve its performance with increasing $p$ due to the nested structure of its OD method. We also conclude that eFD on top of FjORD does not seem to lead to better results. More importantly though, given the heterogeneous pool of devices, to obtain the highest performing model for eFD, multiple models have to be trained (i.e. one per device cluster). For instance, the highest performing models for eFD are $\\mathbf { F } _ { 0 . 4 }$ , $\\mathbf { F } _ { 0 . 6 }$ and ${ \\bf F } _ { 0 . 4 }$ for CIFAR10, FEMNIST and Shakespeare respectively, which can be obtained only a posteriori; after all model variants have been trained. Instead, despite the device heterogeneity, FjORD requires a single training process that leads to a global model that significantly outperforms the best model of eFD (by 2.98 and $2 . 7 3 \\mathrm { p p }$ for CIFAR10 and FEMNIST, respectively, and $0 . 1 3 { \\mathfrak { p } }$ for Shakespeare), while allowing the direct, seamless extraction of submodels due to the nested structure of OD. Empirical evidence of the convergence of FjORD and the corresponding baselines is provided in the Appendix. ",
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+            "Figure 5: Ablation analysis of FjORD with Knowledge Distillation. Ordered Dropout with $D _ { \\mathcal { P } } = \\mathcal { U } _ { 5 }$ , KD - Knowledge distillation. "
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+        "text": "5.3 Ablation Study of KD in FjORD ",
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+        "type": "text",
+        "text": "To evaluate the contribution of our knowledge distillation method to the attainable performance of FjORD, we conduct an ablative analysis on all three datasets. We adopt the same setup of uniform-5 and drop scale $= 1 . 0$ as in the previous section and compare FjORD with and without KD. ",
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+        "type": "text",
+        "text": "Fig. 5 shows the efficacy of FjORD’s KD in $\\mathrm { F L }$ settings. FjORD’s KD consistently improves the performance across all three datasets when $p > 0 . 4$ , with average gains of 0.18, 0.68 and 0.87 pp for submodels of size 0.6, 0.8 and 1 on CIFAR-10, 1.96, 2.39 and 2.65 pp for FEMNIST and $0 . 1 0 \\mathrm { p }$ for Shakespeare. For the cases of $p \\leq 0 . 4$ , the impact of KD is fading. We believe this to be a side-effect of optimising for the average accuracy across submodels, which also yielded the $T = \\alpha = 1$ strategy. We leave the exploration of alternative weighted KD strategies as future work. Overall, the use of KD significantly improves the performance of the global model, yielding gains of 0.71 and $2 . 6 3 \\mathrm { p p }$ for CIFAR10 and FEMNIST and $0 . 1 0 \\mathfrak { p }$ for Shakespeare. ",
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+        "type": "text",
+        "text": "5.4 FjORD’s Deployment Flexibility ",
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+        "type": "text",
+        "text": "5.4.1 Device Clusters Scalability ",
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+        "type": "text",
+        "text": "An important characteristic of FjORD is its ability to scale to a larger number of device clusters or, equivalently, perform well with higher granularity of $p$ values. To illustrate this, we test the performance of OD across two setups, uniform-5 and $- I O$ (defined in $\\ S 5 . 1$ ). ",
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+        "type": "text",
+        "text": "As shown in Fig. 6, FjORD sustains its performance even under the higher granularity of $p$ values. This means that for applications where the modelling of clients needs to be more fine-grained, FjORD can still be of great value, without any significant degradation in achieved accuracy per submodel. This further supports the use-case where device-load needs to be modelled explicitly in device clusters (e.g. modelling device capabilities and load with deciles). ",
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+        "type": "text",
+        "text": "5.4.2 Adaptability to Device Distributions ",
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+        "type": "text",
+        "text": "In this section, we make a similar case about FjORD’s elasticity with respect to the allocation of available devices to each cluster. We adopt the setup of uniform-5 once again, but compare across drop scales 0.5 and 1.0 (defined in $\\ S 5 . 1$ ). In both cases, clients that can support models of $p _ { \\operatorname* { m a x } } ^ { i } \\in \\{ 0 . 2 , \\dots , 0 . 8 \\}$ are equisized, but the former halves the percentage of devices and allocates it to the last (high-end) cluster, now accounting for $60 \\%$ of the devices. The rationale behind this is that the majority of participating devices are able to run the whole original model. ",
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+        "type": "text",
+        "text": "The results depicted in Fig. 7 show that the larger submodels are expectedly more accurate, being updated more often. However, the same graphs also indicate that FjORD does not significantly degrade the accuracy of the smaller submodels in the presence of more high-tier devices (i. $e . \\ d s = 0 . 5$ ). This is a direct consequence of sampling $p$ values during local rounds, instead of tying each tier with only the maximal submodel it can handle. We should also note that we did not alter the uniform sampling in this case on the premise that high-end devices are seen more often, precisely to illustrate FjORD’s adaptability to latent user device distribution changes of which the server may not be aware. ",
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+        "type": "text",
+        "text": "6 Related Work ",
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+        "type": "text",
+        "text": "Dropout Techniques. Contrary to conventional Random Dropout [59], which stochastically drops a different, random set of a layer’s units in every batch and is typically applied for regularisation purposes, OD employs a structured ordered dropping scheme that aims primarily at tunably reducing the computational and memory cost of training and inference. However, OD can still have an implicit regularisation effect since we encourage learning towards the top-ranked units (e.g. the left-most units in the example of Fig. 2), as these units will be dropped less often during training. Respectively, at inference time, the load of a client can be dynamically adjusted by dropping the least important units, i.e. adjusting the width of the network. ",
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+        "image_caption": [
+            "Figure 6: Demonstration of FjORD’s scalability with respect to the number of device clusters. "
+        ],
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+        "image_caption": [
+            "(b) RNN - Shakespeare ",
+            "Figure 7: Demonstration of the adaptability of FjORD across different device distributions. "
+        ],
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+    {
+        "type": "text",
+        "text": "To the best of our knowledge, the only similar technique to OD is Nested Dropout, where the authors proposed a similar construction, which is applied to the representation layer in autoencoders [54] in order to enforce identifiability of the learned representation or the last layer of the feature extractor [24] to learn an ordered set of features for transfer learning. In our case, we apply OD to every layer to elastically adapt the computation and memory requirements during training and inference. ",
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+        "type": "text",
+        "text": "Traditional Pruning. Conventional non-FL compression techniques can be applicable to reduce the network size and computation needs. The majority of pruning methods [18, 16, 37, 35, 49] aim to generate a single pruned model and require access to labelled data in order to perform a costly fine-tuning/calibration for each pruned variant. Instead, FjORD’s Ordered Dropout enables the deterministic extraction of multiple pruned models with varying resource budgets directly after training. In this manner, we remove both the excessive overhead of fine-tuning and the need for labelled data availability, which is crucial for real-world, privacy-aware applications [60, 56]. Finally, other model compression methods [13, 64, 9] remain orthogonal to FjORD. ",
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+        "type": "text",
+        "text": "System Heterogeneity. So far, although substantial effort has been devoted to alleviating the statistical heterogeneity [38] among clients [58, 36, 26, 12, 40], the system heterogeneity has largely remained unaddressed. Considering the diversity of client devices, techniques on client selection [50] and control of the per-round number of participating clients and local iterations [45, 65] have been developed. Nevertheless, as these schemes are restricted to allocate a uniform amount of work to each selected client, they either limit the model complexity to fit the lowest-end devices or exclude slow clients altogether. From an aggregation viewpoint, [39] allows for partial results to be integrated to the global model, thus enabling the allocation of different amounts of work across heterogeneous clients. Despite the fact that each client is allowed to perform a different number of local iterations based on its resources, large models still cannot be accommodated on the more constrained devices. ",
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+    {
+        "type": "text",
+        "text": "Communication Optimisation. The majority of existing work has focused on tackling the communication overhead in FL. [32] proposed using structured and sketched updates to reduce the transmitted data. ATOMO [61] introduced a generalised gradient decomposition and sparsification technique, aiming to reduce the gradient sizes communicated upstream. [17] adaptively select the gradients’ sparsification degree based on the available bandwidth and computational power. Building upon gradient quantisation methods [44, 23, 53, 25], [2] proposed using quantisation in the model sharing and aggregation steps. However, their scheme requires the same clients to participate across all rounds, and is, thus, unsuitable for realistic settings where clients’ availability cannot be guaranteed. Despite the bandwidth savings, these communication-optimising approaches do not offer computational gains nor do they address device heterogeneity. Nonetheless, they remain orthogonal to our work and can be complementarily combined to further alleviate the communication cost. ",
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+        "type": "text",
+        "text": "Computation-Communication Co-optimisation. A few works aim to co-optimise both the computational and bandwidth costs. PruneFL [29] proposes an unstructured pruning method. Despite the similarity to our work in terms of pruning, this method assumes a common pruned model across all clients at a given round, thus not allowing more powerful devices to update more weights. Hence, the pruned model needs to meet the constraints of the least capable devices, which severely limits the model capacity. Moreover, the adopted unstructured sparsity is difficult to translate to processing speed gains [67]. Federated Dropout [8] randomly sparsifies the global model, before sharing it to the clients. Similarly to PruneFL, Federated Dropout does not consider the system diversity and distributes the same model size to all clients. Thus, it is restricted by the low-end devices or excludes them altogether from the FL process. Additionally, Federated Dropout does not translate to computational benefits at inference time, since the whole model is deployed after federated training. ",
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+    {
+        "type": "text",
+        "text": "Contrary to the presented works, our framework embraces the client heterogeneity, instead of treating it as a limitation, and thus pushes the boundaries of FL deployment in terms of fairness, scalability and performance by tailoring the model size to the device at hand, both at training and inference time, in a “train-once-deploy-everywhere” manner. ",
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+        "type": "text",
+        "text": "7 Conclusions & Future Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this work, we have introduced FjORD, a federated learning method for heterogeneous device training. To this direction, FjORD builds on top of our Ordered Dropout technique as a means to extract submodels of smaller footprints from a main model in a way where training the part also participates in training the whole. We show that our Ordered Dropout is equivalent to SVD for linear mappings and demonstrate that FjORD’s performance in the local and federated setting exceeds that of competing techniques, while maintaining flexibility across different environment setups. ",
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+        "type": "text",
+        "text": "In the future, we plan to investigate how FjORD can be deployed and extended to future-gen devices and models in a life-long manner, the interplay between system and data heterogeneity for OD-based personalisation as well as alternative dynamic inference techniques for tackling system heterogeneity. ",
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+    {
+        "type": "text",
+        "text": "Broader Impact ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our work has a dual broader societal impact: i) on privacy and fairness in participation and ii) on the environment. On the one hand, centralised DNN training [19] has been the norm for a long time, mainly facilitated by the advances in server-grade accelerator design and cheap storage. However, this paradigm comes with a set of disadvantages, both in terms of data privacy and energy consumption. With mobile and embedded devices becoming more capable and FL becoming a viable alternative [3, 7], one can leverage the free compute cycles of client devices to train models on-device, without data ever leaving the device premises. These devices, being typically battery-powered, operate under a more constrained power envelope compared to data-center accelerators [1]. Moreover, these devices are already deployed in the wild, but typically not used for training purposes. What FjORD contributes is the ability for even less capable devices to participate in the training process, thus increasing the representation of low-tier devices (and by extension the correlated demographic groups), as well as adding to the overall compute capabilities of the distributed system as a whole, potentially offsetting part of the carbon footprint of centralised training data centers [52]. ",
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+    {
+        "type": "text",
+        "text": "However, moving the computation cost from the service provider to the user of the device is a non-negligible step and the user should be made aware what their device is used for, especially if they are contributing to the knowledge of a model they do not own. Moreover, while many large data centers [11, 15] are increasingly dependent on renewable resources for meeting their power demands, this might not be the case for household electricity, which may impede the sustainability of training on device, at least in the short run. ",
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+    {
+        "type": "text",
+        "text": "Funding Disclosure ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This work was entirely performed at and funded by Samsung AI. ",
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+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
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+        "type": "text",
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parse/train/BJl6AjC5F7/BJl6AjC5F7.md

@@ -0,0 +1,415 @@
+# LEARNING TO REPRESENT EDITS
+
+Pengcheng Yin∗, Graham Neubig Language Technology Institute Carnegie Mellon University Pittsburgh, PA 15213, USA {pcyin,gneubig}@cs.cmu.edu
+
+# Miltiadis Allamanis, Marc Brockschmidt, Alexander L. Gaunt
+
+Microsoft Research Cambridge, CB1 2FB, United Kingdom {miallama,mabrocks,algaunt}@microsoft.com
+
+# ABSTRACT
+
+We introduce the problem of learning distributed representations of edits. By combining a “neural editor” with an “edit encoder”, our models learn to represent the salient information of an edit and can be used to apply edits to new inputs. We experiment on natural language and source code edit data. Our evaluation yields promising results that suggest that our neural network models learn to capture the structure and semantics of edits. We hope that this interesting task and data source will inspire other researchers to work further on this problem.
+
+# 1 INTRODUCTION
+
+One great advantage of electronic storage of documents is the ease with which we can edit them, and edits are performed in a wide variety of contents. For example, right before a conference deadline, papers worldwide are finalized and polished, often involving common fixes for grammar, clarity and style. Would it be possible to automatically extract rules from these common edits? Similarly, program source code is constantly changed to implement new features, follow best practices and fix bugs. With the widespread deployment of (implicit) version control systems, these edits are quickly archived, creating a major data stream that we can learn from.
+
+In this work, we study the problem of learning distributed representations of edits. We only look at small edits with simple semantics that are more likely to appear often and do not consider larger edits; i.e., we consider “add definite articles” rather than “rewrite act 2, scene 3.” Concretely, we focus on two questions: i) Can we group semantically equivalent edits together, so that we can automatically recognize common edit patterns? ii) Can we automatically transfer edits from one context to another? A solution to the first question would yield a practical tool for copy editors and programmers alike, automatically identifying the most common changes. By leveraging tools from program synthesis, such groups of edits could be turned into interpretable rules and scripts (Rolim et al., 2017). When there is no simple hard rule explaining how to apply an edit, an answer to the second question would be of great use, e.g., to automatically rewrite natural language following some stylistic rule.
+
+We propose to handle edit data in an autoencoder-style framework, in which an “edit encoder” $f _ { \Delta }$ is trained to compute a representation of an edit ${ \pmb x } _ { - }  { \pmb x } _ { + }$ , and a “neural editor” $\alpha$ is trained to construct $\mathbf { \pmb { x } } _ { + }$ from the edit representation and ${ \bf x } _ { - }$ . This framework ensures that the edit representation is semantically meaningful, and a sufficiently strong neural editor allows this representation to not be specific to the changed element. We experiment with various neural architectures that can learn to represent and apply edits and hope to direct the attention of the research community to this new and interesting data source, leading to better datasets and stronger models.
+
+Briefly, the contributions of our paper are: (a) in Sect. 2, we present a new and important machine learning task on learning representations of edits (b) we present a family of models that capture the structure of edits and compute efficient representations in Sect. 3 (c) we create a new source code edit dataset, and release the data extraction code at https://github.com/Microsoft/msrc-dpu-learning-to-represent-edits and the data at http://www.cs.cmu.edu/˜pengchey/githubedits.zip. (d) we perform a set of experiments on the learned edit representations in Sect. 4 for natural language text and source code and present promising empirical evidence that our models succeed in capturing the semantics of edits.
+
+![](images/8d21428d5873234cdcd2a3a846d04788112a4f4b1f1086b00d37d61a8564e4d4.jpg)  
+Figure 1: Given an edit (Edit 1) of ${ \bf x } _ { - }$ to $\mathbf { \pmb { x } } _ { + }$ , $f _ { \Delta }$ computes an edit representation vector. Using that representation vector the neural editor $\alpha$ applies the same edit to a new $\mathbf { { x } } _ { - } ^ { \prime }$ . The code snippets shown here are real code change examples from the roslyn open-source compiler project.
+
+# 2 TASK
+
+In this work, we are interested in learning to represent and apply edits on discrete sequential or structured data, such as text or source code parse trees1. Figure 1 gives a graphical overview of the task, described precisely below.
+
+Edit Representation Given a dataset of edits $\{ \mathbf { x } _ { - } ^ { ( i ) }  \mathbf { x } _ { + } ^ { ( i ) } \} _ { i = 1 } ^ { N }$ , where $\pmb { x } _ { - } ^ { ( i ) }$ is the original version of some object and $\pmb { x } _ { + } ^ { ( i ) }$ its edited form (see upper half of Figure 1 for an example), our goal is to learn a representation function $f _ { \Delta }$ that maps an edit operation ${ \pmb x } _ { - }  { \pmb x } _ { + }$ to a real-valued edit representation $f _ { \Delta } ( \pmb { x } _ { - } , \pmb { x } _ { + } ) \in \mathbb { R } ^ { n }$ . A desired quality of $f _ { \Delta }$ is for the computed edit representations to have the property that semantically similar edits have nearby representations in $\mathbb { R } ^ { n }$ . Having distributed representations also allows other interesting downstream tasks, e.g., unsupervised clustering and visualization of similar edits from large-scale data (e.g. the GitHub commit stream), which would be useful for developing human-assistance toolkits for discovering and extracting emerging edit patterns (e.g. new bug fixes or emerging “best practices” of coding).
+
+Neural Editor Given an edit representation function $f _ { \Delta }$ , we want to learn to apply edits in a new context. This can be achieved by learning a neural editor $\alpha$ that accepts an edit representation $f _ { \Delta } ( { \pmb x } _ { - } , { \pmb x } _ { + } )$ and a new input $\mathbf { x } _ { - } ^ { \prime }$ and generates $\pmb { x } _ { + } ^ { \prime }$ .2 This is illustrated in the lower half of Figure 1.
+
+# 3 MODEL
+
+We cast the edit representation problem as an autoencoding task, where we aim to minimize the reconstruction error of $\alpha$ for the edited version $\mathbf { \pmb { x } } _ { + }$ given the edit representation $f _ { \Delta } ( { \pmb x } _ { - } , { \pmb x } _ { + } )$ and the original version ${ \bf x } _ { - }$ . By limiting the capacity of $f _ { \Delta }$ ’s output and allowing the model to freely use information about ${ \bf x } _ { - }$ , we are introducing a “bottleneck” that forces the overall framework to not simply treat $f _ { \Delta } ( { \pmb x } _ { - } , { \pmb x } _ { + } )$ as an encoder of $\mathbf { \pmb { x } } _ { + }$ . The main difference from traditional autoencoders is that in our setup, an optimal solution requires to re-use as much information as possible from ${ \bf x } _ { - }$ to make the most of the capaneural network and a dataset $f _ { \Delta }$ ly, given a probabilistic editor function , we seek to minimize the negative likel $P _ { \alpha }$ such as aod loss $\{ \pmb { x } _ { - } ^ { ( i ) }  \pmb { x } _ { + } ^ { ( i ) } \} _ { i = 1 } ^ { N }$
+
+![](images/e7efd8f1e1fb21d94866addf08fd7c5d752432d562b6fda6496edfb792ef9393.jpg)  
+Figure 2: (a) Graph representation of statement ${ \texttt { u } } = { \texttt { x } } + { \texttt { x } }$ . Rectangular (resp. rounded) nodes denote tokens (resp. non-terminals). (b) Sequence of tree decoding steps yielding $\mathrm { ~  ~ x ~ } + \mathrm { ~  ~ x ~ } - 2 3$ , where $\mathrm { ~  ~ { ~ x ~ } ~ } + \mathrm { ~  ~ { ~ x ~ } ~ }$ is copied (using the TREECP action) from the context graph in (a).
+
+$$
+\mathcal { L } = - \frac { 1 } { N } \sum _ { i } \log P _ { \alpha } ( \pmb { x } _ { + } \mid \pmb { x } _ { - } , f _ { \Delta } ( \pmb { x } _ { - } , \pmb { x } _ { + } ) ) .
+$$
+
+Note that this loss function can be interpreted in two ways: (1) as a conditional autoencoder that encodes the salient information of an edit, given ${ \bf x } _ { - }$ and (2) as an encoder-decoder model that encodes ${ \bf x } _ { - }$ and decodes $\mathbf { \delta } _ { \mathbf { x } _ { + } }$ conditioned on the edit representation $f _ { \Delta } ( { \pmb x } _ { - } , { \pmb x } _ { + } )$ . In the rest of this section, we discuss our methods to model $P _ { \alpha }$ and $f _ { \Delta }$ as neural networks.
+
+# 3.1 NEURAL EDITOR
+
+As discussed above, $\alpha$ should use as much information as possible from ${ \bf x } _ { - }$ , and hence, an encoderdecoder architecture with the ability to copy from the input is most appropriate. As we are primarily interested in edits on text and source code in this work, we explored two architectures: a sequenceto-sequence model for text, and a graph-to-tree model for source code, whose known semantics we can leverage both on the encoder as well as on the decoder side. Other classes of edits, for example, image manipulation, would most likely be better served by convolutional neural models.
+
+Sequence-to-Sequence Neural Editor First, we consider a standard sequence-to-sequence model with attention (over the tokens of ${ \bf x } _ { - }$ ). The architecture of our sequence-to-sequence model is similar to that of Luong et al. (2015), with the difference that we use a bidirectional LSTM in the encoder and a token-level copying mechanism (Vinyals et al., 2015) that directly copies tokens into the decoded sequence. Whereas in standard sequence-to-sequence models the decoder is initialized with the representation computed by the encoder, we initialize it with the concatenation of encoder output and the edit representation. We also feed the edit representation as input to the decoder LSTM at each decoding time step. This allows the LSTM decoder to take the edit representation into consideration while generating the output sequence.
+
+Graph-to-Tree Neural Editor Our second model aims to take advantage of the additional structure of ${ \pmb x } _ { - }$ and $\mathbf { \pmb { x } } _ { + }$ . To achieve this, we combine a graph-based encoder with a tree-based decoder. We use $T ( \pmb { x } )$ to denote a tree representation of an element, e.g., the abstract syntax tree (AST) of a fragment of source code. We extend $T ( { \pmb x } )$ into a graph form $G ( \pmb { x } )$ by encoding additional relationships (e.g., the “next token” relationship between terminal nodes, etc.) (see Figure 2(a)). To encode the elements of $G ( { \pmb x } _ { - } )$ into vector representations, we use a gated graph neural network (GGNN) (Li et al., 2015). Similarly to recurrent neural networks for sequences (such as biRNNs), GGNNs compute a representation for each node in the graph, which can be used in the attention mechanisms of a decoder. Additionally, we use them to obtain a representation of the full input ${ \bf x } _ { - }$ , by computing their weighted average following the strategy of Gilmer et al. (2017) (i.e., computing a score for each node, normalizing scores with a softmax, and using the resulting values as weights).
+
+Our tree decoder follows the semantic parsing model of Yin & Neubig (2018), which sequentially generate a tree $T ( \pmb { x } _ { + } )$ as a series of expansion actions $a _ { 1 } \ldots a _ { N }$ . The probability of taking an action is modeled as $p ( a _ { t } \mid a _ { < t } , s )$ , where $s$ is the input (a sequence of words in the original semantic parsing setting) and $a _ { < t }$ is the partial tree that has been generated so far. The model of Yin & Neubig (2018) mainly uses two types of actions: EXPANDR expands the current non-terminal using a grammar rule, and GENTERM generates a terminal token from a vocabulary or copies a token from $s ^ { 3 }$ . The dependence on the partial tree $a _ { < t }$ is modeled by an LSTM cell which is used to maintain state throughout the generation procedure. Additionally, the LSTM receives the decoder state used to pick the action at the parent node as an additional input (“parent-feeding”). This process illustrated in Figure 2(b).
+
+![](images/cb39fb0e7c75647fcc65334fd303ed17c7e744415f2df8a569903205d9a687fa.jpg)  
+Figure 3: Sequence (a) and graph (b) representation of edit of $\mathrm { ~ v ~ . ~ F ~ } = \mathrm { ~ x ~ + ~ \ x ~ t o ~ u ~ = ~ x ~ + ~ \ x ~ }$
+
+We extend this model to our setting by replacing the input sequence $s$ by ${ \bf x } _ { - }$ ; concretely, we condition the decoder on the graph-level representation computed for $G ( { \pmb x } _ { - } )$ . Additionally, we use the change representation $f _ { \Delta } ( \cdot )$ as an additional input to the LSTM initial state and at every decoding step. Based on the observation that edits to source code often manipulate the syntax tree by moving expressions around (e.g. by nesting statements in a conditional, or renaming a function while keeping its arguments), we extend the decoding model of Yin & Neubig (2018) by adding a facility to copy entire subtrees from the input. For this, we add a decoder action TREECP. This action is similar to standard copying mechanism known from pointer networks (Vinyals et al., 2015), but instead of copying only a single token, it copies the whole subtree pointed to.
+
+However, adding the TREECP action means that there are many correct generation sequences for a target tree. This problem appears in token-copying as well, but can be easily circumvented by marginalizing over all correct choices at each generation step (by normalizing the probability distribution over allowed actions to sum up those that have the same effect). In the subtree-copying setting, the lengths of action sequences representing different choices may differ. In our implementation we handle this problem during training by simpling picking the generation sequence that greedily selecting TREECP actions.
+
+# 3.2 EDIT REPRESENTATION
+
+To compute a useful edit representation, a model needs to focus on the differences between ${ \bf x } _ { - }$ and $\mathbf { \pmb { x } } _ { + }$ . A risk in our framework is that $f _ { \Delta }$ degenerates into an encoder for $\mathbf { \pmb { x } } _ { + }$ , turning $\alpha$ into a decoder. To avoid this, we need to follow the standard autoencoder trick, i.e. it is important to limit the capacity of the result of $f _ { \Delta }$ by generating the edit representation in a low-dimensional space $\mathbb { R } ^ { N }$ . This acts as a bottleneck and encodes only the information that is needed to reconstruct $\mathbf { \pmb { x } } _ { + }$ from ${ \bf x } _ { - }$ . We again experimented with both sequence-based and graph-based representations of edits.
+
+Sequence Encoding of Edits Given x− (resp. x+) as sequence of tokens t(0)− , . . . t(T−)− (resp. $t _ { + } ^ { ( 0 ) } , \ldots t _ { + } ^ { ( T _ { + } ) } )$ . t(T+)+ ), we can use a standard (deterministic) diffing algorithm to compute an alignment of tokens in the two sequences. We then use extra symbols for padding, for additions, − for deletions, for replacements, and $=$ for unchanged tokens to generate a single sequence representing both ${ \bf x } _ { - }$ and $\mathbf { \pmb { x } } _ { + }$ . This is illustrated in Figure 3(a). By embedding the three entries in each element of the sequence separately and concatenating their representation, they can be fed into a standard sequence encoder whose final state is our desired edit representation. In this work, we use a biLSTM.
+
+Graph Encoding of Edits As in the graph-to-tree neural editor, we represent ${ \bf x } _ { - }$ and $\mathbf { \pmb { x } } _ { + }$ as trees $T ( { \pmb x } _ { - } )$ and $T ( \pmb { x } _ { + } )$ . We combine these trees into a graph representation $G ( \pmb { x } _ { - }  \pmb { x } _ { + } )$ by merging both trees into one graph, using “Removed”, “Added” and “Replaced” edges. To connect the two trees, we compute the same alignment as in the sequence case, connecting leaves that are the same and each replaced leaf to its replacement. We also propagate this information up in the trees, i.e., two inner nodes are connected by $\cdot \underline { { \epsilon } } = \ '$ edges if all their descendants are connected by $\cdot \underline { { \epsilon } } = \ '$ edges. This is illustrated in Figure 3(b). Finally, we also use the same ${ ^ { 6 4 } + ^ { 9 3 } } / { ^ { 6 4 } - ^ { 3 3 } } / { ^ { 6 4 } } \{  ^ { 3 3 }  / { ^ { 6 4 } = ^ { 3 3 } }$ tags for the initial node representation, computing it as the concatenation of the string label (i.e. token or nonterminal name) and the embedding of the tag. To obtain an edit representation, we use a GGNN unrolled for a fixed number of timesteps and again use the weighted averaging strategy of Gilmer et al. (2017).
+
+# 4 EVALUATION
+
+Evaluating an unsupervised representation learning method is challenging, especially for a newly defined task. Here, we aim to evaluate the quality of the learned edit representations with a series of qualitative and quantitative metrics on natural language and source code.
+
+# 4.1 DATASETS AND CONFIGURATION
+
+Natural Language Edits We use the WikiAtomicEdits (Faruqui et al., 2018) dataset of pairs of short edits on Wikipedia articles. We sampled $1 0 4 0 K$ edits from the English insertion portion of the dataset and split the samples into $1 0 0 0 K / 2 0 K / 2 0 K$ train-valid-test sets.
+
+Source Code Edits To obtain a dataset for source code, we clone a set of $5 4 \mathrm { C } \#$ projects on GitHub and collected a GitHubEdits dataset (see Appendix A for more information). We selected all changes in the projects that are no more than 3 lines long and whose surrounding 3 lines of code before and after the edited lines have not been changed, ensuring that the edits are separate and short. We then parsed the two versions of the source code and take as ${ \bf x } _ { - }$ and $\mathbf { \pmb { x } } _ { + }$ the code that belongs to the top-most AST node that contains the edited lines. Finally, we remove trivial changes such variable renaming, changes within comments or formatting changes. Overall, this yields 111 724 edit samples. For each edit we run a simple C# analysis to detect all variables and normalize variable names such that each unique variable within ${ \bf x } _ { - }$ and $\mathbf { \pmb { x } } _ { + }$ has a unique normalized name V0, V1, etc. This step is necessary to avoid the sparsity of data induced by the variety of different identifier naming schemes. We split the dataset into $9 1 , 3 7 2 / 1 0 , 1 7 6 / 1 0 , 1 7 6$ samples as train/valid/test sets.
+
+Additionally, we introduce a labeled dataset of source code edits by using C# “fixers”. Fixers are small tools built on top of the C# compiler, used to perform common refactoring and modernization tasks (e.g., using new syntactic sugar). We selected 16 of these fixers and ran them on 6 C# projects to generate a small C#Fixers dataset of 2,878 edit pairs with known semantics. We present descriptions and examples of each fixer in Appendix A.
+
+Configuration Throughout the evaluation we use a fixed size of 512 for edit representations. The size of word embeddings and hidden states of encoding LSTMs is 128. The dimensionality of the decoding LSTM is set to 256. Details of model configuration can be found in Sect. A.
+
+When generating the target $\mathbf { \pmb { x } } _ { + }$ , our neural editor model can optionally take as input the context of the original input ${ \bf x } _ { - }$ (e.g., the preceding and succeeding code segments surrounding $\mathbf { \delta _ { x _ { - } } }$ ), whose information could be useful for predicting $\mathbf { \pmb { x } } _ { + }$ . For example, in source code edits the updated code snippet $\mathbf { \pmb { x } } _ { + }$ may reuse variables defined in the preceding snippet. In our code experiments, we use a standard bidirectional LSTM network to encode the tokenized 3 lines of code before and after ${ \bf x } _ { - }$ as context. The encoded context is used to initialize the decoder, and as an additional source for the pointer network to copy tokens from.
+
+# 4.2 QUALITY OF EDIT REPRESENTATIONS
+
+First, we study the ability of our models to encode edits in a semantically meaningful way.
+
+Visualizing Edits on Fixers Data In a first experiment, we train our sequential neural editor model on our GitHubEdits data and then compute representations for the edits generated by the C# fixers. A $\mathbf { t } { - } \mathbf { S N E }$ visualization (Maaten & Hinton, 2008) of the encodings is shown in Figure 4. For this visualization, we randomly selected 100 examples from the edits of each fixer (if that fixer has more than 100 samples) and discarded fixer categories with less than 40 examples. Readers are referred to Appendix A for detailed descriptions of each fixer category. We find that our model produces dense clusters for simple or distinctive code edits, e.g. fixer RCS1089 (using the $^ { + + }$ or -- unary operators instead of a binary operator (e.g., $\mathrm { ~ i ~ } = \mathrm { ~ i ~ } + \mathrm { ~ 1 ~ }  \mathrm { ~ i + + } )$ , and fixer $\mathrm { C A } 2 0 0 7$ (adding .ConfigureAwait(false) for await statements). We also analyzed cases where (1) the edit examples from the same fixer
+
+![](images/3d44a6ec0b93f1baf6d11a9b7848e7e29d81ae5e78124e919ac758fa978f0ddd.jpg)  
+Figure 4: t-SNE visualization of edits from 13 $\mathbf { C } \#$ fixers, where point color indicates the fixer. Labels indicate the id of the fixer, see main text.
+
+are scattered, or (2) the clusters of different fixers overlap with each other. For example, the fixer RCS1077 covers 12 different aspects of optimizing LINQ method calls (e.g., type casting, counting, etc.), and hence its edits are scattered. On the other hand, fixers $\mathtt { R C S 1 1 4 6 }$ and RCS1206 yield overlapping clusters, as both fixers change code to use the ?. operator. Fixers RCS1207 (change a lambda to a method group, e.g. foo $( \mathrm { x } { = } > \mathrm { b a r } ( \mathrm { x } ) )  \mathrm { f o o } ( \mathrm { b a r } )$ ) and RCS1021 (simplify lambda expressions, e.g. foo $( \mathrm { x } { = } > \left\{ \mathrm { r e t u r n ~ } ~ 4 \div \left. \right\} ) ~ \right. ~ \mathrm { f o o } ~ ( \mathrm { x } { = } > 4 ) ~ .$ ) are similar, as both inline lambda expressions in two different ways. Analysis yields that the representation is highly dependent on surface tokens. For instance, IDE004 (removing redundant type casts, e.g. (int) $2  2$ ) and RCS1207 (removing explicit argument lists) yield overlapping clusters, as both involve deleting identifiers wrapped by parentheses.
+
+Human Evaluation on Encoding Natural Language WikiAtomicEdits In a second experiment, we test how well neighborhoods in edit representation space correspond to semantic similarity. We computed the five nearest neighbors of 200 randomly sampled seed edits from our training set, using both our trained sequence-to-sequence editing model with sequential edit encoder, as well as a simple bag-of-words baseline based on TF-IDF scores. We then rated the quality of the retrieved neighbors on a scale of 0 (“unrelated edit”), 1 (“similar edit”) and 2 (“semantically or syntactically same edit”). Details of the annotation schema is included in Sect. E. We show the (normalized) discounted cumulative gain (DCG, Manning et al. (2008)) for the two models at the top of Tab. 1 (higher is better). The relevance scores indicate that our neural model clearly outperforms the simplistic baseline. Tab. 1 also presents two example edits with their nearest neighbors. Example 1 shows that the neural edit models succeeded in representing syntactically and semantically similar edits, while the bag-of-words baseline relies purely on surface token overlap. Interestingly, we also observed that the edit representations learned by the neural editing model on WikiAtomicEdits are somewhat sensitive to position, i.e. the position of the inserted tokens in both the seed edit and the nearest neighbors is similar. This is illustrated in Example 2, where the second (“senegalese striker”) and the third (“republican incumbent”) nearest neighbors returned by the neural model have similar editing positions as the seed edit, while they are semantically diverse.
+
+# 4.3 EDIT ENCODER PERFORMANCE
+
+To evaluate the performance of our two edit encoders discussed in Sect. 3.2 and disentangle it from the choice of neural editor, we train various combinations of our neural editor model and manually evaluate the quality of the edit representation. More specifically, we trained our neural editor models on GitHubEdits and randomly sampled 200 seed edits and computed their 3 nearest neighbors using each end-to-end model. We then rated the resulting groups using the same 0-2 scale as above. The resulting relevance scores are shown in Tab. 2.
+
+<table><tr><td colspan="2">Bag of Words Model</td><td>Seq2Seq - Seq Edit Encoder</td></tr><tr><td>DCG/NDCG@5</td><td>9.3 / 67.3%</td><td>13.5 / 90.3%</td></tr><tr><td>DCG@5 (by edit size)</td><td>1: 14.7 2-3: 10.8 &gt;3: 5.4</td><td>1: 16.2 2-3: 12.9 &gt;3: 12.4</td></tr></table>
+
+Table 1: Natural language human evaluation results and 3 nearest neighbors. IInserted textJ marked. Example 1 neural editing model returns syntactically and semantically similar edits. Example 2 Neural edit representations are sensitive to position.   
+
+<table><tr><td colspan="2">DCG/NDCG@5 9.3 / 67.3% 13.5 / 90.3% DCG@5 (by edit size) 1: 14.7 2-3:10.8 &gt;3: 5.4 1: 16.2 2-3:12.9 &gt;3:12.4</td></tr><tr><td colspan="2">Example1 daniel james nava(born february 22,1983)is anamericanprofessional baseball outfielder nava is only the fourth player inmlb historyto hit a grand slam in his frst major league at bat and the second to</td></tr><tr><td>do it on the first pitch . NN-1 outfielder.</td><td>he batted .3O2 with 73 steals，and received a ?arthurraybriles(borndecember3,1955) september call -up to the major leaguesas an isaformer american football coach andhis most recent head coaching position was at bay- lor university,a position he held from the 2008</td></tr><tr><td>NN-2 he played as an outfielderfor the hanshin tigers.</td><td>season through the 2015 season . ?jonathan david disalvatore（bornmarch 30 ,1981）isaprofessional icehockeyhewas selected by the san jose sharks in the 4th round</td></tr><tr><td>NN-3 in 2012,his senior at oak mountain,dahl had a .412 batting average,34 runs batted in（rbis）, and 18 stolen bases as an outfielder.</td><td>(104th overall）of the 2OOO nhl entry draft ■professor paul talalay(born march 31,1923 ）is the john jacob abeldistinguished service professor of pharmacology and director of the laboratory for molecular sciences at johns hop- kins school of medicine in baltimore .</td></tr><tr><td colspan="2">Example2 she,along with her follow artist carolyn mase studied with impressionist landscape painter john henry twachtman at the art students league of new york . his brother was draughtsman william daniell the first painting was a portrait of a young girl</td></tr><tr><td>NN-1 daniell.</td><td>and his uncle was?landscape painterthomas ,emerantia van beresteyn,the sister of the landscape painternicolaes van beresteyn,the laterfounder of half of this hofje .</td></tr><tr><td>NN-2</td><td>william james linton（december7,1812 - de- he was the club &#x27;s top scorer with 22 goals in cember 29,1897）was an english - born ameri- all competitions,one more than?senegalese canwood engraver,?landscape painter，po- strikerlamine diarra,who left the club at the litical reformer and author of memoirs,novels， end of the season .</td></tr><tr><td>poetry and non-fiction NN-3 early on,hoppermodeled his style after chase and french?impressionist masters douard manet and edgar degas .</td><td>caforio ”aggressively attacked”his opponent, ?republican incumbentsteve knight,for his delayed response to the leak.</td></tr></table>
+
+Table 2: Relevance scores of human evaluation on GitHubEdits data. Acc. $@ 1$ denotes the ratio that the 1-nearest neighbor has a score 2.   
+
+<table><tr><td>Model</td><td>DCG@3</td><td>NDCG@3 (%)</td><td>Acc.@1(%)</td></tr><tr><td>BoW</td><td>7.77</td><td>75.99</td><td>58.46</td></tr><tr><td>Seq2Seq - Seq Edit Encoder</td><td>10.09</td><td>90.05</td><td>75.90</td></tr><tr><td>Graph2Tree -Seq Edit Encoder</td><td>10.56</td><td>91.40</td><td>79.49</td></tr><tr><td>Graph2Tree-Graph Edit Encoder</td><td>9.44</td><td>86.20</td><td>72.31</td></tr></table>
+
+Comparing the sequential edit encoders trained with Seq2Seq and Graph2Tree editors, we found that the edit encoder trained with the Graph2Tree objective performs better. We hypothesize that this is because the Graph2Tree editor better captures structural-level information about an edit. For instance, Example 1 in Tab. 3 removes explicit type casting. The Seq2Seq editor has difficulty distinguishing this type of edit, confusing it with changes of lambda expressions to method groups (1st and 2nd nearest neighbors) since both two types of edits involve removing paired parentheses.
+
+Surprisingly, we found that the graph-based edit encoder does not outperform the sequence-based encoder. However, we observe that the graph edit encoder in many cases tends to better capture high-level and abstract structural edit patterns. Example 2 in Tab. 3 showcases a seed edit that swaps two consecutive declarations, which corresponds to swapping the intermediate Expression nodes representing each statement on the underlying AST. In this case, the graph edit encoder is capable of grouping semantically similar edits, while it seems to be more difficult for the sequential encoder encoder to capture the edit pattern. On the other hand, we found that the graph edit encoder often fails to capture simpler, lexical level edits (e.g., Example 1). This might suggest that terminal node information is not effectively propagated, an interesting issue worth future investigation.
+
+Table 3: Two example source code edits and their nearest neighbors based on the edit representations computed by each model.   
+
+<table><tr><td colspan="2">Example 1 Example 2</td></tr><tr><td>x_:vo.SendSelectSoundRequest((int) V1); x+:v0.SendSelectSoundRequest (v1);</td><td>x_:string vO； string V1; x+:string V1; string vO;</td></tr><tr><td>Seq2Seq-Seq Edit Encoder</td><td>Seq2Seq-Seq Edit Encoder</td></tr><tr><td>? x_:VO.Debug((）=&gt; LITERAL); x+:VO.Debug(LITERAL);</td><td>? x_:RetryConfig V0; string V1; x+:string V1;RetryConfig V0;</td></tr><tr><td>1 x_:VO.Debug((） =&gt; LITERAL); x+:VO.Debug(LITERAL);</td><td>? x_: string[] vO; string[] V1;int V2; x+:int V2; string[] VO; string[] V1;</td></tr><tr><td>x_:V0.WriteCompressedInteger((uint)Vl);</td><td>x_:Type VO= null; BindingFlags V1= 0;</td></tr><tr><td>x+:v0.WriteCompressedInteger(v1); Graph2Tree-Seq Edit Encoder</td><td>x+:BindingFlags Vl= 0; Type VO= null;</td></tr><tr><td colspan="2">Graph2Tree-Seq Edit Encoder x_:V0.WriteCompressedInteger((uint)Vl);</td></tr><tr><td>x+:V0.WriteCompressedInteger(Vl);</td><td>x_:RetryConfig V0; string V1; x+:string Vl; RetryConfig V0;</td></tr><tr><td>x_:vo.WriteCompressedInteger((uint)V1);</td><td>x_:string[] V0; string[] Vl;int V2;</td></tr><tr><td>x+:V0.WriteCompressedInteger(v1);</td><td>x+:int V2; string[] VO; string[] V1;</td></tr><tr><td>x_:Vo.WriteCompressedInteger((uint)V1); x+:V0.WriteCompressedInteger(Vl);</td><td>? x_:int VO = V1;int V2 = V3; x+:int V2 = V3;int VO = V1;</td></tr><tr><td colspan="2">Graph2Tree-Graph Edit Encoder Graph2Tree-Graph Edit Encoder</td></tr><tr><td>x_:V0.UpdateLastRead(this.V1); ? x+:VO.UpdateLastRead(V1);</td><td>? x_:RetryConfig V0; string V1;</td></tr><tr><td>x_:V0.UpdateLastWrite(this.V1);</td><td>x+: string Vl;RetryConfig V0;</td></tr><tr><td>x+:VO.UpdateLastWrite(Vl);</td><td>? x_:int V0 = V1;int V2 = V3; x+:int V2 = V3; int VO = V1;</td></tr><tr><td>x_:vo.Append(this.V1);</td><td> x_: double V0= -1; double V1= -1;</td></tr></table>
+
+Table 4: Test performance of different neural editors.   
+
+<table><tr><td>Model</td><td>Acc.@1 (%)</td><td>Recall@5 (%)</td><td>PPL per token</td></tr><tr><td>GitHubEdits</td><td></td><td></td><td></td></tr><tr><td>Seq2Seq-Bag-of-Edits Encoder</td><td>44.05</td><td>54.97</td><td>1.4808</td></tr><tr><td>Seq2Seq -Seq Edit Encoder</td><td>59.63</td><td>65.46</td><td>1.2792</td></tr><tr><td>Graph2Tree-Bag-of-Edits Encoder</td><td>40.66</td><td>49.42</td><td>1.5058</td></tr><tr><td>Graph2Tree -Seq Edit Encoder</td><td>57.49</td><td>62.94</td><td>1.3043</td></tr><tr><td>Graph2Tree -Graph Edit Encoder</td><td>48.05</td><td>56.51</td><td>1.3712</td></tr><tr><td>WikiAtomicEdits</td><td></td><td></td><td></td></tr><tr><td>Seq2Seq-Bag-of-Edits Encoder</td><td>23.73</td><td>43.47</td><td>1.3730</td></tr><tr><td>Seq2Seq - Seq Edit Encoder</td><td>72.94</td><td>76.53</td><td>1.0527</td></tr></table>
+
+# 4.4 PRECISION OF NEURAL EDITORS
+
+Finally, we evaluate the performance of our end-to-end system by predicting the edited input $\mathbf { \pmb { x } } _ { + }$ given ${ \pmb x } _ { - }$ and the edit representation. We are interested in answering two research questions: First, how well can our neural editors generate $\mathbf { \pmb { x } } _ { + }$ given the gold-standard edit representation $f _ { \Delta } ( { \pmb x } _ { - } , { \pmb x } _ { + } ) \ P$ Second, and perhaps more interestingly, can we use the representation of a similar edit $f _ { \Delta } ( \pmb { x } _ { - } ^ { \prime } , \pmb { x } _ { + } ^ { \prime } )$ to generate $\mathbf { \pmb { x } } _ { + }$ by applying that edit to ${ \bf x } _ { - }$ (i.e. $\hat { \pmb x } _ { + } = \alpha ( { \pmb x } _ { - } , f _ { \Delta } ( { \pmb x } _ { - } ^ { \prime } , { \pmb x } _ { + } ^ { \prime } ) ) ) _  \} ^ { \epsilon }$
+
+To answer the first question, we trained our neural editor models on the WikiAtomicEdits and the GitHubEdits dataset, and evaluate the performance of encoding and applying edits on test sets. For completeness, we also evaluated the performance of our neural editor models with a simple “Bag-ofEdits” edit encoding scheme, where $f _ { \Delta } ( { \pmb x } _ { - } , { \pmb x } _ { + } )$ is modeled as the concatenation of two vectors, each representing the sum of the embeddings of added and deleted tokens in the edit, respectively. This edit encoding method is reminiscent of the model used in Guu et al. (2017) for solving a different task of language modeling by marginalizing over latent edits, which we will elaborate in Sect. 5. Tab. 4 lists the evaluation results. With our proposed sequence- and graph-based edit encoders, our neural editor models achieve reasonable end-to-end performance, surpassing systems using bag-of-edits representations. This is because many edits are context-sensitive and position-sensitive, requiring edit representation models that go beyond the bag-of-edits scheme to capture those effects (more analysis is included in Appendix B). Interestingly, on the GitHubEdits dataset, we find that the Seq2Seq editor with sequential edit encoder registers the best performance. However, it should be noted that in this set of experiments, we encode the gold-standard edit $f _ { \Delta } ( { \pmb x } _ { - } , { \pmb x } _ { + } )$ to predict $\mathbf { \boldsymbol { x } } _ { + }$ . As we will show later, better performance with the gold-standard edit does not necessarily imply better (more generalizable) edit representation. Nevertheless, we hypothesize that the higher accuracy of the Seq2Seq edit is due to the fact that a significant proportion of edits in this dataset is small and primarily syntactically simple. Indeed we find that $69 \%$ of test examples have a token-level edit distance of less than 5.
+
+Table 5: Transfer learning results on C# fixers data, averaged across all fixer categories.   
+
+<table><tr><td>Model</td><td>Acc.(%)</td><td>Acc.*(%)</td><td>Recall@5(%)</td><td>Recall@5*(%)</td></tr><tr><td>Seq2Seq - Seq Edit Encoder</td><td>38.35</td><td>77.67</td><td>41.50</td><td>83.84</td></tr><tr><td>Graph2Tree -Seq Edit Encoder</td><td>49.21</td><td>77.30</td><td>51.93</td><td>81.77</td></tr><tr><td>Baselines (no edit encoding)</td><td>7.07</td><td></td><td>14.29</td><td></td></tr><tr><td>Seq2Seq w/o Edit Encoder</td><td>8.81</td><td></td><td>11.90</td><td></td></tr><tr><td>Graph2Tree w/o Edit Encoder</td><td></td><td></td><td></td><td></td></tr></table>
+
+∗: upper-bound performance of predicting $\mathbf { x } _ { + }$ using the gold-standard edit representations.
+
+To answer the second question, we use the trained neural editors from the previous experiment, and test their performance in a “one-shot” transfer learning scenario. Specifically, we use our high-quality C#Fixers dataset, and for each fixer category $\mathcal { F }$ of semantically similar edits, we randomly select a seed edit $\{ \pmb { x } _ { - } ^ { \prime }  \pmb { x } _ { + } ^ { \prime } \} \in \mathcal { F }$ , and use its edit representation $\bar { f } _ { \Delta } ( \pmb { x } _ { - } ^ { \prime } , \pmb { x } _ { + } ^ { \prime } )$ to predict the updated code for all examples in $\mathcal { F }$ , i.e., we have $\hat { \pmb { x } } _ { + } = \alpha ( \pmb { x } _ { - } , f _ { \Delta } ( \pmb { x } _ { - } ^ { \prime } , \pmb { x } _ { + } ^ { \prime } ) ) , \forall \{ \pmb { x } _ { - }  \pmb { x } _ { + } \} \in \mathcal { F }$ . This task is highly non-trivial, since a fixer category could contain more than hundreds of edit examples collected from different $\mathbf { C } \#$ projects. Therefore, it requires the edit representations to generalize and transfer well, while being invariant of local lexical information like specific method names. To make the experimental evaluation more robust to noise, for each fixer category $\mathcal { F }$ , we randomly sample 10 seed edit pairs $\{ \pmb { x } _ { - } ^ { \prime }  \pmb { x } _ { + } ^ { \prime } \}$ , compute their edit representations and use them to predict the edited version of the examples in $\mathcal { F }$ and evaluate accuracy of predicting the exact final version. We then report the best score among the 10 seed representations as the performance metric on $\mathcal { F }$ .
+
+Tab. 5 summarizes the results and also reports the upper bound performance when using the goldstandard edit representation $f _ { \Delta } ( { \pmb x } _ { - } , { \pmb x } _ { + } )$ to predict $\mathbf { \delta } _ { \mathbf { x } _ { + } }$ , and an approximation of the “lower bound” accuracies using pre-trained Seq2Seq and Graph2Tree models without edit encoders. We found that our neural Graph2Tree editor with the sequential edit encoder significantly outperforms the Seq2Seq editor, even though Seq2Seq performs better when using gold-standard edit representations. This suggest that the edit representations learned with the Graph2Tree model generalize better, especially for edits discussed in Sect. 4.2 that involve syntactic variations like RCS1021 (lambda expression simplification, $7 . 8 \%$ vs. $3 0 . 7 \%$ for Seq2Seq and Graph2Tree), and RCS1207 (change lambdas to method groups, $7 . 1 \%$ vs. $2 6 . 2 \%$ ). Interestingly, we also observe that Seq2Seq outperforms the Graph2Tree model for edits with trivial surface edit sequences, where the Graph2Tree model does not have a clear advantage. For example, on RCS1015 (use nameof operator, e.g. Exception $( " \mathrm { x } " )  \mathrm { E x c } $ eption(nameof $\mathbf { \Psi } ( \mathbf { x } )$ )), the accuracies for Seq2Seq and Graph2Tree are $4 0 . 0 \%$ (14/35) and $2 8 . 6 \%$ (10/35), resp. We include more analysis of the results in Appendix C.
+
+# 5 RELATED WORK
+
+Edits have recently been considered in NLP, as they represent interesting linguistic phenomena in language modeling and discourse (Faruqui et al., 2018; Yang et al., 2017a). Specifically, Guu et al. (2017) present a generative model of natural language sentences via editing prototypes. Our work shares with Guu et al. (2017) in that (1) the posterior edit encoding model in Guu et al. (2017) is similar to our baseline “bag-of-edits” encoder in Sec. 4.4, and (2) the sequence-to-sequence sentence generation model given the prototype and edit representation is reminiscent of our Seq2Seq editor. In contrast, our work directly focuses on discriminative learning of representing edits and applying the learned edits for both sequential (NL) and structured (code) data. Another similar line of research is “retrieve-and-edit” models for text generation (Hashimoto et al., 2018), where given an input data $_ { \textbf { \em x } }$ , the target prediction $\textbf {  { y } }$ is generated by editing a similar target $\boldsymbol { y } ^ { \prime }$ that is retrieved based on the similarity between its source $\mathbf { { x } ^ { \prime } }$ and the input $_ { \textbf { \em x } }$ . While these models typically require an “editor” component to generate the output by exploiting the difference between similar inputs, they usually use the simpler bag-of-edits representations (Wu et al., 2019), or implicitly capture it via end-to-end neural networks (Contractor et al., 2018). To our best knowledge, there is not any related work that classifies or otherwise explicitly represents the differences over similar input, with the exception of differential recurrent neural networks used for action recognition in videos (Veeriah et al., 2015; Zhuang et al., 2018). This is a substantially different task, as the data includes a temporal component as well.
+
+Source code edits are a widely studied artifact. Specialized software, such as git, is widely used to store source code revision histories. Nguyen et al. (2013) studied the repetitiveness of source code changes by identifying identical types of changes using a deterministic differencing tool. In contrast, we employ on a neural network to cluster similar changes together. Rolim et al. (2017) use such clusters to synthesize small programs that perform the edit. The approach is based on Rolim et al. (2018) extract manually designed syntactic features from code and cluster over multiple changes to find repeatable edit rules. Similarly, Paletov et al. (2018) extract syntactic features specifically targeting edits in cryptography API protocols. In this work, we try to avoid hand-designed features and allow a neural network to learn the relevant aspects of a change by directly giving as input the original and final version of a changed code snippet.
+
+Modeling tree generation with machine learning is an old problem that has been widely studied in NLP. Starting with Maddison & Tarlow (2014), code generation has also been considered as a tree generation problem. Close to our work is the decoder of Yin & Neubig (2017) which we use as the basis of our decoder. The work of Chen et al. (2018) is also related, since it provides a tree-to-tree model, but focuses on learning a single translation tasks and cannot be used directly to represent multiple types of edits. Both Yin & Neubig (2017) and Chen et al. (2018) have copying mechanism for single tokens, but our subtree copying mechanism is novel.
+
+Autoencoders (see Goodfellow et al. (2016) for an overview) have a long history in machine learning. Variational autoencoders (Kingma & Welling, 2013) are similar to autoencoders but instead of focusing on the learned representation, they aim to create accurate generative probabilistic models. Most (variational) autoencoders focus on encoding images but there have been works that autoencode sequences, such as text (Dai & Le, 2015; Bowman et al., 2015; Yang et al., 2017b) and graphs (Simonovsky & Komodakis, 2018; Liu et al., 2018). Conditional variational autoencoders (Sohn et al., 2015) have a related form to our model (with the exception of the KL term), but are studied as generative models, whereas we are primarily interested in the edit representation.
+
+# 6 DISCUSSION & CONCLUSIONS
+
+In this work, we presented the problem of learning distributed representation of edits. We believe that the dataset of edits is highly relevant and should be studied in more detail. While we have presented a set of initial models and metrics on the problem and obtained some first promising results, further development in both of these areas is needed. We hope that our work inspires others to work on this interesting problem in the future.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Rachel Free for her insightful comments and suggestions.
+
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+
+Martin Simonovsky and Nikos Komodakis. GraphVAE: Towards generation of small graphs using variational autoencoders. arXiv preprint arXiv:1802.03480, 2018.
+
+Kihyuk Sohn, Honglak Lee, and Xinchen Yan. Learning structured output representation using deep conditional generative models. In Advances in Neural Information Processing Systems, pp. 3483–3491, 2015.
+
+Vivek Veeriah, Naifan Zhuang, and Guo-Jun Qi. Differential recurrent neural networks for action recognition. In Proceedings of the IEEE international conference on computer vision, pp. 4041– 4049, 2015.
+
+Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. In Advances in Neural Information Processing Systems, pp. 2692–2700, 2015.
+
+Yu Ping Wu, Furu Wei, Shaohan Huang, Zhoujun Li, and Ming Zhou. Response generation by context-aware prototype editing. 2019.
+
+Diyi Yang, Aaron Halfaker, Robert Kraut, and Eduard Hovy. Identifying semantic edit intentions from revisions in wikipedia. In Proceedings of the Conference on Empirical Methods in Natural Language Processing, pp. 2000–2010, 2017a.
+
+Zichao Yang, Zhiting Hu, Ruslan Salakhutdinov, and Taylor Berg-Kirkpatrick. Improved variational autoencoders for text modeling using dilated convolutions. arXiv preprint arXiv:1702.08139, 2017b.
+
+Pengcheng Yin and Graham Neubig. A syntactic neural model for general-purpose code generation. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), volume 1, pp. 440–450, 2017.
+
+Pengcheng Yin and Graham Neubig. TRANX: A transition-based neural abstract syntax parser for semantic parsing and code generation. In Conference on Empirical Methods in Natural Language Processing (EMNLP) Demo Track, 2018.
+
+Naifan Zhuang, The Duc Kieu, Guo-Jun Qi, and Kien A Hua. Deep differential recurrent neural networks. arXiv preprint arXiv:1804.04192, 2018.
+
+# A DATASETS AND CONFIGURATION
+
+WikiAtomicEdits We randomly sampled $1 0 4 0 K$ insertion examples from the English portion of WikiAtomicEdits (Faruqui et al., 2018) dataset, with a train, development and test splits of $1 0 0 0 K$ , $2 0 K$ and $2 0 K$ .
+
+GitHubEdits We cloned the top $5 4 \mathrm { C } \#$ GitHub repositories based on their popularity (Tab. 8). For each commit in the master branch, we collect the previous and updated versions of the source code, and extract all consecutive lines of edits that are smaller than three lines, and with at least three preceding and successive lines that have not been changed. We then filter trivial changes such as variable and identifier renaming, and changes happened within comments. We also limit the number of tokens for each edit to be smaller than 100, and down-sample edits whose frequency is larger than 30. Finally, we split the dataset by commit ids, ensuring that there are no edits in the training and testing (development) sets coming from the same commit. Tab. 6 lists some statistics of the dataset.
+
+Table 6: Statistics of the GitHubEdits Dataset   
+
+<table><tr><td>Average Num. Tokens in x_ Average Num. Tokens in x+ Average Edit Distance Average size of AST for T(x_) Average size of AST for T(x+)</td><td>16.4 17.0 5.0 28.5 29.4</td></tr></table>
+
+C#Fixers We selected $1 6 \ \mathrm { ~ C \# ~ }$ fixers from Roslyn4 and Roslynator5, and ran them on 6 C# projects to generate a small, high-quality $\mathbf { C } \#$ fixers dataset of 2 878 edit pairs with known semantics. Table 7 lists the detailed descriptions for each fixer category. And more information can be found at https://github.com/JosefPihrt/Roslynator/blob/master/ src/Analyzers/README.md.
+
+Network Configuration Throughout the experiments, we use a fixed edit representation size of 512. The dimensionality of word embedding, the hidden states of the encoder LSTMs, as well as the gated graph neural network is 128, while the decoder LSTM uses a larger hidden size of 256. For the graph-based edit encoder, we used a two-layer graph neural network, with 5 information propagation steps at each layer. During training, we performed early stopping, and choose the best model based on perplexity scores on development set. During testing, we decode a target element $\mathbf { \pmb { x } } _ { + }$ using a beam size of 5.
+
+Table 7: Descriptions of fixer categories in C#Fixers dataset   
+
+<table><tr><td>Fixer ID</td><td>Description</td><td>Num. Edits</td><td>Example</td><td></td></tr><tr><td>CA2007</td><td>apply .ConfigureAwait(false) to await statements</td><td>1051</td><td>x_:await x+:await</td><td>Console.WriteAsync() Console.WriteAsync() .ConfigureAwait(false)</td></tr><tr><td>IDE0004</td><td>Cast is redundant</td><td>53</td><td>x-:var X= x+:var X=</td><td>1； var b= (int)x; 1； var b=x;</td></tr><tr><td>RCS1015</td><td>Use nameof operator</td><td>35</td><td></td><td>x_:Exception(&quot;parameter&quot;); x+:Exception(nameof(parameter));</td></tr><tr><td>RCS1021</td><td>Simplify lambda expression</td><td>411</td><td>m_: }）；</td><td>var x = items.Select(f =&gt; return f.ToString();</td></tr><tr><td></td><td></td><td></td><td>var x+:</td><td>x = items.Select( f =&gt; f.ToString();</td></tr><tr><td>RCS1032</td><td> Remove redundant parentheses</td><td>24</td><td>x-:if x_:if （x）{</td><td>（(x））{}</td></tr><tr><td>RCS1058</td><td>Use compound assignment</td><td>43</td><td>c_:i = x+:i +=2；</td><td>i+2；</td></tr><tr><td>RCS1077</td><td>Optimize LINQ method call Use --/++ operator instead of</td><td>200</td><td></td><td>x-:items.Where(f =&gt; Foo(f)).Any(); x+:items.Any(f =&gt; Foo(f));</td></tr><tr><td>RCS1089</td><td>assignment</td><td>75</td><td>x_:i +=1； x+:i</td><td>=i+1; x_:var x = s.ToString();</td></tr><tr><td></td><td>Remove redundant ToSt ring call</td><td>20</td><td>x+:var . string x_:</td><td>x= s; &quot;a&quot;； s=</td></tr><tr><td></td><td></td><td></td><td>string const string s = x+: string s2</td><td>s2 2=s+ &quot;b&quot;; &quot;a&quot;; =s+&quot;b&quot;；</td></tr><tr><td>RCS1123</td><td>Add parentheses according to operator precedence</td><td>109</td><td>x_:if (x 二 x+:if (x 二</td><td>Y &amp;&amp;z) { （y&amp;&amp; z））{}</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>x+:items?.First().ToString();</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>RCS1207</td><td>Use method group instead of anonymous function</td><td>42</td><td></td><td>x+: int i = x?.GetHashCode() ?? 0; x_:items.Select(f =&gt; Foo(f));</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>x-:sb.Append(s + &quot;x&quot;);</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>RCS1146</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>Use conditional access</td><td>71</td><td>x_:x != null</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>&amp;&amp; x.StartsWith(&quot;a&quot;);</td></tr><tr><td></td><td>Optimize call of St ringBuilder&#x27;s</td><td></td><td></td><td> X+:x?.StartsWith(&quot;a&quot;);</td></tr><tr><td>RCS1197</td><td></td><td>95</td><td></td><td></td></tr><tr><td></td><td>Append/AppendLine</td><td></td><td></td><td>x+: sb.Append(s) .Append(&quot;x&quot;);</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>RCS1202</td><td>Avoid NullReferenceException</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>56</td><td></td><td>x_:items.First().ToString();</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>RCS1206</td><td>Use conditional access instead of</td><td></td><td></td><td>int i = (x != null) ？</td></tr><tr><td></td><td></td><td>116</td><td>x_:</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>conditional expression</td><td></td><td></td><td> x.Value.GetHashCode() : 0;</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr></table>
+
+Table 8: Our C# GitHub dataset projects   
+
+<table><tr><td>Name</td><td>GitHub Id</td><td>Description</td></tr><tr><td>acat</td><td>intel/acat</td><td>Assistive Context-Aware Toolkit</td></tr><tr><td>akka.net</td><td>akka/akka.net</td><td>Distributed Actors</td></tr><tr><td>aspnetboilerplate</td><td>aspnetboilerplate/aspnetboilerplate</td><td>ASP.NET boilerplate</td></tr><tr><td>AutoMapper</td><td>AutoMaper/AutoMapper</td><td>Object-Object Mapper</td></tr><tr><td>BotBuilder</td><td>Microsoft/BotBuilder</td><td>Bot Framework</td></tr><tr><td>CefSharp</td><td>cefsharp/CefSharp</td><td>Chromium Embedded Framework Bindings</td></tr><tr><td>choco</td><td>chocolatey/choco</td><td>package mananger</td></tr><tr><td>cli</td><td>dotnet/cli</td><td>.NET CLI Tools</td></tr><tr><td>CodeHub</td><td>CodeHubApp/CodeHub</td><td>iOS application</td></tr><tr><td>coreclr</td><td>dotnet/coreclr</td><td>.NETFramework</td></tr><tr><td>corefx</td><td>dotnet/corefx</td><td>.NET FOundational Libraries</td></tr><tr><td>dapper</td><td>StackExchange/Dapper</td><td>Object Mapper</td></tr><tr><td>dnSpy</td><td>0xd4d/dnSpy</td><td>.NET debugger and assembly editor</td></tr><tr><td>duplicati</td><td>duplicati/duplicati</td><td>Encrypted Cloud Backups</td></tr><tr><td>EntityFramework</td><td>aspnet/EntityFramework</td><td>Object-Relational Mapper</td></tr><tr><td>EntityFrameworkCore</td><td>aspnet/EntityFrameworkCore</td><td>Object-Relational Mapper-Core</td></tr><tr><td>FluentValidation</td><td>JeremySkinner/FluentValidation</td><td>Validation Rules</td></tr><tr><td>framework</td><td>accord-net/framework</td><td>ML,CV Framework</td></tr><tr><td>GVFS</td><td>Microsoft/VFSForGit</td><td>Git Virual File System</td></tr><tr><td>Hangfire</td><td>HangfireIO/Hangfire</td><td>Background job library</td></tr><tr><td>ILSpy</td><td>icsharpcode/ILSpy</td><td>Decompiler</td></tr><tr><td>JavaScriptServices</td><td>aspnet/JavaScriptServices</td><td>ASP.NETJS Services</td></tr><tr><td>MahApps.Metro</td><td>MahApps/MahApps.Metro</td><td>WPFFramework</td></tr><tr><td>MaterialDesignInXamlToolkit</td><td>MaterialDesignInXamlToolkit/Materi-</td><td>DesignXAML&amp;WPF</td></tr><tr><td>mono</td><td>alDesignInXamlToolkit mono/mono</td><td>.NET implementation</td></tr><tr><td>monodevelop</td><td>mono/monodevelop</td><td>IDE</td></tr><tr><td>MonoGame</td><td>MonoGame/MonoGame</td><td>Game Framework</td></tr><tr><td>msbuild</td><td>Microsoft/msbuild</td><td>Build Tool</td></tr><tr><td>Mvc</td><td>aspnet/Mvc</td><td>MVCFramework</td></tr><tr><td>Nancy</td><td>NancyFx/Nancy</td><td>HTTP based services</td></tr><tr><td>Newtonsoft.Json</td><td>JamesNK/Newtonsoft.Json</td><td>JSON framework</td></tr><tr><td>NLog</td><td>NLog/NLog</td><td>Loggin for.NET</td></tr><tr><td>OpenLiveWriter</td><td>OpenLiveWriter/ OpenLiveWriter</td><td>Text editor</td></tr><tr><td>OpenRA</td><td>OpenRA/OpenRA</td><td>Strategy Game Engine</td></tr><tr><td>Opserver</td><td>opserver/Opserver</td><td>Monitoring System</td></tr><tr><td>orleans</td><td>dotnet/orleans</td><td>Distributed Virtual Actors</td></tr><tr><td>PowerShell</td><td>PowerShell/PowerShell</td><td>Command Line</td></tr><tr><td>Psychson</td><td>brandonlw/Psychson</td><td>Firmware</td></tr><tr><td>PushSharp</td><td>Redth/PushSharp</td><td>Push Notifications</td></tr><tr><td>ravendb</td><td>ravendb/ravendb</td><td>Database</td></tr><tr><td>ReactiveUI</td><td>reactiveui/ReactiveUI</td><td>Reactive MVC Framework</td></tr><tr><td>RestSharp</td><td>restsharp/RestSharp</td><td>HTTP/REST Client</td></tr><tr><td>roslyn</td><td>dotnet/roslyn</td><td>.NET Compiler</td></tr><tr><td>Rx.NET</td><td>dotnet/reactive</td><td>Reactive extensions.</td></tr><tr><td>ServiceStack</td><td>ServiceStack/ServiceStack</td><td>Web Service Framework</td></tr><tr><td>shadowsocks-windows</td><td></td><td>Cryptography</td></tr><tr><td></td><td>shadowsocks/ shadowsocks-windows</td><td>Screen Recorder</td></tr><tr><td>ShareX</td><td>ShareX/ShareX</td><td>Real-time web framework</td></tr><tr><td>SignalR</td><td>SignalR/SignalR Sonarr/Sonarr</td><td>PVR</td></tr><tr><td>Sonarr</td><td>KeenSoftwareHouse/ SpaceEngineers</td><td>Game</td></tr><tr><td>SpaceEngineers SparkleShare</td><td>hbons/SparkleShare</td><td>File Sharing</td></tr><tr><td>StackExchange.Redis</td><td>StackExchange/StackExchange.Redis</td><td>Redis Client</td></tr><tr><td>WaveFunctionCollapse</td><td>mxgmn/WaveFunctionCollapse</td><td>Bitmap/tilemap Generator</td></tr><tr><td>Wox</td><td>Wox-launcher/Wox</td><td>Launcher</td></tr></table>
+
+# B CLUSTERING EXPERIMENTS
+
+To qualitatively evaluate the quality of the learned edit representations. We use the models trained on the WikiAtomicEdits and GitHubEdits datasets to cluster natural language and code edits. We run K-Means clustering algorithm on 0.5 million sampled edits from WikiAtomicEdits, and all $9 0 K$ code edits from GitHubEdits, producing 50 000 and 20 000 clusters for each dataset.
+
+Tab. 9 and Tab. 10 list some example clusters on WikiAtomicEdits and GitHub datasets, respectively. Due to the size of clusters, we omit out-liners and present distinctive examples from each cluster. On the WikiAtomicEdits dataset, we found clusters whose examples are semantically and syntactically similar. More interestingly, on the source code data, we find representative clusters that relate to idiomatic patterns and best practices of programming. The clustering results produced by our model would be useful for programming synthesis toolkits to generate interpretable code refractory rules, which we leave as interesting future work.
+
+Finally, we remark that the clustering results indicate that the encoding of edits is context-sensitive and position-sensitive for both natural language and source code data. For instance, the WikiAtomicEdits examples we present in Tab. 9 clearly indicate that semantically similar insertions also share similar editing positions. This is even more visible in code edits (Tab. 10). For instance, in the first example in Tab. 10, Equal() can be changed to Empty() only in the Assert namespace (i.e., the context). These examples demonstrate that it is important for an edit encoder to capture the contextual and positional information in edits, a property that cannot be captured by simple “bag-of-edits” edit representation methods.
+
+Description Add a person’s middle name
+
+1. isaiah ImarcusJ rankin ( born 22 may 1978 in london ) is an english professional footballer currently playing for stevenage borough .   
+2. audrey IkathleenJ brown ( born 24 may , 1913 ) is a british athlete who competed mainly in the 100 metres .   
+3. alice IedithJ rumph was a painter , etcher , and teacher .   
+4. mark IlarryJ taufua is an australian professional rugby league player .   
+5. monique IedithJ lamoureux ( born july 3 , 1989 ) is an american ice hockey player .   
+1. mid-state regional airport I, also known as mid-state airport ,J is a small airport on in rush township , centre county in pennsylvania in the united states .   
+2. islamic culture I, also known as saracenic culture ,J is a term primarily used in secular academia to describe the cultural practices common to historically islamic peoples .   
+3. birds of prey I, also known as raptors ,J are birds that hunt for food primarily via flight , using their keen senses , especially vision .   
+4. tetyana styazhkina I, also written as tetyana stiajkina ,J ( ; born april 10 , 1977 ) is a ukrainian cycle racer who rides for the chirio forno d’asolo team .   
+5. acid jazz I, also known as club jazz ,J is a musical genre that combines elements of jazz , soul , funk , disco and hip hop .   
+1. the douro fully enters portuguese territory just after the confluence with the gueda river ; once the douro enters portugal , major population centres are less frequent Ialong the riverJ .   
+2. mochou lake and mochou lake park are located at 35 hanzhongmen da jie in the jianye district of nanjing , china I, west to qinhuai riverJ   
+3. reiner gamma is an albedo feature that is located on the oceanus procellarum , to the west of the reiner crater Ion the moonJ   
+4. she made a brief return to the screen in ” parrish ” ( 1961 ) , playing the supporting role of mother which received little attention Iby the pressJ .   
+5. he was involved in a few storylines , including one where he broke his toe and had a heart attack after he was pushed by a mugger Iin the marketJ .
+
+Description Add positional or temporal clause
+
+1. Iat the timeJ ajax and hercules were trapped behind a landslide at the gaillard cut , both were working to clear the landslide .   
+2. Iat the docks ,J hikaru attempts to befriend the tiger , but finds that it dislikes humans .   
+3. Iabout the second ,J i do know they exist , but the question is whether they are considered a genre outside of japan .   
+4. Iin the battle ,J shirou uses his reality marble , unlimited blade works and defeats gilgamesh .   
+5. Iin the game ,J red is a curious 11 - year - old boy from pallet town . $\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ Type ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ == null ? null : V1.GetType();   
+$\mathbf { \pmb { x } } _ { + }$ Type V0 = V1?.GetType();   
+$\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ $\begin{array} { r l } { { \nabla } 0 } & { { } = } \end{array}$ ( $( \mathbb { V } \mathbb { 1 } \mathbb { 1 } =$ null) ? V1.Operand : null) as MemberExpression;   
+$\mathbf { \pmb { x } } _ { + }$ $\mathrm { V } 0 ~ = ~ \mathrm { V } 1 ? . 0$ perand as MemberExpression;   
+$\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ string ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1 ~ = =$ null ? null : V1.GetType().Name;   
+x+ string $\begin{array} { r } { \nabla 0 = \mathrm { ~ \nabla ~ } \mathrm { V } 1 \boldsymbol { ? } } \end{array}$ .GetType().Name;   
+$\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ == null ? null : V1(V2).ToArray();   
+$^ { \mathbf { \alpha } _ { \mathbf { \mathcal { X } } + \mathbf { \beta } } }$ var V0 = V1?.Invoke(V2).ToArray();
+
+# Description Optimize LINQ queries
+
+var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ .Customers.Where( ${ \begin{array} { r l } { \ V { 2 } } & { { } = > } \end{array} }$ V2.CustomerID $= =$ LITERAL)   
+$\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ .FirstOrDefault();   
+$\mathbf { \pmb { x } } _ { + }$ var V0 $=$ V1.Customers .FirstOrDefault $( \nabla 2 \ ) \ = > \ \nabla 2$ .CustomerID $= =$ LITERAL);   
+$\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ .TypeConverters.Where(V2 $= >$ V2.CanConvertTo(V3, V1)) .FirstOrDefault();   
+x+ var V0 $=$ V1.TypeConverters .FirstOrDefault(V2 $= >$ V2.CanConvertTo(V3, V1));   
+$\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ var V0 $=$ this.V1.Where( ${ \begin{array} { r l } { \ V { 2 } } & { { } = > } \end{array} }$ V2.CanDeserialize(V3)) .FirstOrDefault();   
+$^ { \mathbf { \alpha } _ { \mathbf { \mathcal { X } } + \mathbf { \beta } } }$ var $\begin{array} { r l } { { \nabla } 0 } & { { } = } \end{array}$ this.V1.FirstOrDefault( $^ { ' } \mathrm { V } 2 \ = \mathrm { > }$ V2.CanDeserialize(V3));   
+$\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ var V0 $=$ V1.Where( ${ \begin{array} { r l } { \ V { 2 } } & { { } = > } \end{array} }$ V2.Item1 $\scriptstyle = = \ V 3$ && V2.Item2 $\scriptstyle \ = \ \ = \ \lor 4$ ) .FirstOrDefault();   
+x+ var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ .FirstOrDefault( $\mathrm { ~ \textit ~ { ~ V ~ 2 ~ } ~ } = >$ V2.Item1 $\scriptstyle = = \ \nabla 3$ && V2.Item2 == V4);
+
+Description Change from Add function to indexer.
+
+$\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ V0.Add(V1.key, V1.V2);   
+$^ { \mathbf { \alpha } _ { \mathbf { \mathcal { X } } + \mathbf { \beta } } }$ V0[V1.key] $=$ V1.V2;   
+$\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ V0.Add(V1.Id, V2);   
+$\mathbf { \pmb { x } } _ { + }$ V0[V1.Id] = V2;   
+$\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ V0.Add(V1.Etag, V1);   
+$\mathbf { \pmb { x } } _ { + }$ V0[V1.Etag] = V1;   
+$\mathbf { _ { 2 } } \mathbf { _ { 2 } }$ V0.Add(V1.V2, V3.Merge(V1.V4));   
+$^ { \mathbf { \alpha } _ { \mathbf { \mathcal { X } } + \mathbf { \beta } } }$ V0[V1.V2] = V3.Merge(V1.V4);
+
+Table 11: Break-down performance results on the transfer learning task. See Tab. 7 for descriptions of each fixer category.   
+
+<table><tr><td colspan="6">Graph2Tree Seq Edit Encoder</td><td colspan="3">Seq2Seq Seq Edit Encoder</td></tr><tr><td>Fixer ID</td><td>Acc.(%)</td><td>Acc.*(%)</td><td>Recall@5(%)</td><td>Recall@5*(%)</td><td>Acc.(%)</td><td>Acc.*(%)</td><td>Recall@5(%)</td><td>Recall@5*(%)</td></tr><tr><td>CA2007</td><td>88.0</td><td>89.2</td><td>88.2</td><td>94.3</td><td>52.7</td><td>91.9</td><td>61.0</td><td>93.8</td></tr><tr><td>IDE0004</td><td>69.8</td><td>92.5</td><td>73.6</td><td>94.3</td><td>45.3</td><td>98.1</td><td>45.3</td><td>98.1</td></tr><tr><td>RCS1015</td><td>28.6</td><td>82.9</td><td>40.0</td><td>82.9</td><td>40.0</td><td>71.4</td><td>42.9</td><td>71.4</td></tr><tr><td>RCS1021</td><td>30.7</td><td>60.8</td><td>33.3</td><td>67.6</td><td>7.8</td><td>56.2</td><td>17.8</td><td>72.3</td></tr><tr><td>RCS1032</td><td>8.3</td><td>37.5</td><td>8.3</td><td>45.8</td><td>20.8</td><td>45.8</td><td>20.8</td><td>45.8</td></tr><tr><td>RCS1058</td><td>93.0</td><td>88.4</td><td>95.3</td><td>90.7</td><td>37.2</td><td>69.8</td><td>39.5</td><td>76.7</td></tr><tr><td>RCS1077</td><td>6.5</td><td>69.5</td><td>6.5</td><td>74.0</td><td>7.5</td><td>84.0</td><td>7.5</td><td>84.5</td></tr><tr><td>RCS1089</td><td>96.0</td><td>98.7</td><td>98.7</td><td>98.7</td><td>76.0</td><td>98.7</td><td>76.0</td><td>98.7</td></tr><tr><td>RCS1097</td><td>15.0</td><td>90.0</td><td>15.0</td><td>90.0</td><td>25.0</td><td>90.0</td><td>25.0</td><td>95.0</td></tr><tr><td>RCS1118</td><td>95.4</td><td>98.1</td><td>99.6</td><td>99.6</td><td>93.7</td><td>99.6</td><td>98.7</td><td>1.00</td></tr><tr><td>RCS1123</td><td>66.1</td><td>81.7</td><td>68.8</td><td>86.2</td><td>64.2</td><td>87.2</td><td>65.1</td><td>94.5</td></tr><tr><td>RCS1146</td><td>54.9</td><td>81.7</td><td>56.3</td><td>85.9</td><td>45.1</td><td>76.1</td><td>57.7</td><td>91.5</td></tr><tr><td>RCS1197</td><td>5.3</td><td>25.3</td><td>5.3</td><td>33.7</td><td>12.6</td><td>40.0</td><td>12.6</td><td>50.0</td></tr><tr><td>RCS1202</td><td>28.6</td><td>67.9</td><td>37.5</td><td>75.0</td><td>28.6</td><td>69.6</td><td>32.1</td><td>80.4</td></tr><tr><td>RCS1206</td><td>75.0</td><td>99.1</td><td>75.9</td><td>99.1</td><td>50.0</td><td>1.00</td><td>50.0</td><td>1.00</td></tr><tr><td>RCS1207</td><td>26.2</td><td>73.8</td><td>28.6</td><td>90.5</td><td>7.1</td><td>64.3</td><td>11.9</td><td>88.1</td></tr></table>
+
+∗: upper-bound performance of predicting $\mathbf { x } _ { + }$ using the gold-standard edit representations.
+
+# C BREAK-DOWN ANALYSIS OF TRANSFER LEARNING RESULTS
+
+Tab. 11 lists the detailed evaluation results for the transfer learning experiments discussed in Sect. 4.4. We refer readers to Tab. 7 for detailed descriptions of each fixer category. The neural Graph2Tree editor outperforms the Seq2Seq editor (both with sequential edit encoders) on 10 out of 16 fixer categories. However, we found that there are categories where both end-to-end system under-performs, even though the upper-bound accuracy is high (e.g. RCS1077, RCSRCS1197, RCS1207, RCS1032). While improving the generalization ability of the neural editor models to achieve better transfer learning performance is an important future work, we remark that this task is indeed non-trivial. First, some fixer categories cover a broad range of similar edits, which could not be captured by a single seed edit. xSecond, some categories contain syntactically or semantically complex refactoring rules. For instance, RCS1207 converts method groups into anonymous functions, involving changing multiple positions of the source code, which might not be trivially captured by the sequential edit encoder from a single example edit. Additionally, RCS1197 requires reasoning about a chain of expressions. It turns sb.Append $( \ s 1 \ + \ \ s 2 \ + \ \ldots \ + \ \ s \mathrm { N } )$ ) into sb.Append(s1).Append(s2).[. . .]Append(sN)), which our current models are unable to reason about. More interestingly, we found that there are cases where the edits are syntactically simple, but could be semantically more difficult to learn. For instance, RCS1032 is about removing redundant parentheses from expressions. Although the edit pattern might seem to be syntactically simple at the AST level (replacing a ParethesizedExpressionSyntax node by its child node), determining which pair of parentheses is actually redundant in an expression (e.g. $( \ a + \ b )$ ) $\star \mathrm { ~  ~ { ~      ~ } ~ } / \mathrm { ~  ~ { ~ d ~ } ~ } )$ ) is semantically non-trivial to learn from a single edit example. We believe that further advances in (general) learning from source code are required to correctly handle these cases.
+
+# D IMPACT OF TRAINING SET SIZE
+
+To evaluate the data efficiency of our proposed approach, we tested the end-to-end performance of our neural editor model (Sect. 4.4, Tab. 4) with varying amount of training data. Tab. 12 lists the results. We found both Graph2Tree and Seq2Seq editors are relatively data efficient. They registered around $9 0 \%$ of the accuracies achieved using the full training set with only $6 0 \%$ of the training data.
+
+# E DETAILS OF HUMAN EVALUATION
+
+As discussed in Sect. 4.2, we performed human evaluation to rate the qualities of neighboring edits given a seed edit. The annotation instructions on GithubEdits and WikiAtomicEdits datasets are listed below. The annotation was carried out by three authors of this paper, and we anonymized the source of systems that generated the output. The three-way Fleiss’ kappa inter-rater agreement is $\kappa = 0 . 5 5$ , which shows moderate agreement (Artstein & Poesio, 2008), an agreement level that is also used in other annotation tasks in NLP (Faruqui & Das, 2018).
+
+Table 12: Test performance of end-to-end experiments with varying amount of training data.   
+
+<table><tr><td>Training Set Size</td><td>Acc.@1(%)</td><td>Recall@5 (%)</td><td>PPL per token</td></tr><tr><td colspan="2">GitHubEdits</td><td></td><td></td></tr><tr><td colspan="2">Graph2Tree-Seq Edit Encoder</td><td></td><td></td></tr><tr><td>20%</td><td>43.88</td><td>50.53</td><td>1.5703</td></tr><tr><td>40%</td><td>50.44</td><td>56.63</td><td>1.4152</td></tr><tr><td>60%</td><td>53.78</td><td>60.00</td><td>1.3720</td></tr><tr><td>80%</td><td>55.51</td><td>60.85</td><td>1.3392</td></tr><tr><td>100%</td><td>57.49</td><td>62.94</td><td>1.3043</td></tr><tr><td colspan="2">WikiAtomicEdits</td><td></td><td></td></tr><tr><td colspan="2">Seq2Seq-Seq Edit Encoder</td><td></td><td></td></tr><tr><td>20%</td><td>42.87</td><td>48.24</td><td>1.4123</td></tr><tr><td>40%</td><td>57.72</td><td>62.31</td><td>1.1812</td></tr><tr><td>60%</td><td>65.22</td><td>69.62</td><td>1.1070</td></tr><tr><td>80%</td><td>68.44</td><td>73.34</td><td>1.0751</td></tr><tr><td>100%</td><td>72.94</td><td>76.53</td><td>1.0527</td></tr></table>
+
+Table 13: Annotation Instruction for GitHubEdits Data   
+
+<table><tr><td>Rating 2 Semantically and Syntactically Equivalent</td></tr><tr><td>The changed constituents in the seed edit and the neighboring edit are applied to the similar positions of the original sentence, serving the same syntactic and semantic role. For example,</td></tr></table>
+
+# Examples
+
+# • Seed Edit
+
+${ \bf x } _ { - }$ var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ .Where( ${ \begin{array} { r l } { \ V { 2 } } & { { } = > } \end{array} }$ V2.Name $= =$ LITERAL).Single();   
+$\mathbf { \pmb { x } } _ { + }$ var V0 $=$ V1.Single( ${ \bar { \mathsf { V } } } { \bar { \mathsf { Z } } } { = } { > }$ V2.Name $= =$ LITERAL);
+
+# • Neighbor
+
+${ \bf x } _ { - }$ var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ .GetMembers().Where(V2 $= >$ V2.Kind $= =$ SymbolKind.Property).Single();  
+$\mathbf { \pmb { x } } _ { + }$ var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ .GetMembers().Single(V2 $= >$ V2.Kind $= =$ SymbolKind.Property);
+
+# • Seed Edit
+
+${ \bf x } _ { - }$ Type $\mathrm { ~ \small ~ \displaystyle ~ V 0 ~ } ~ = ~ \mathrm { ~ \small ~ V 1 ~ } ~ = =$ null ? typeof(object) : V1.GetType();   
+$\mathbf { \pmb { x } } _ { + }$ Type V0 $=$ V1?.GetType() ?? typeof(object);
+
+# • Neighbor
+
+${ \bf x } _ { - }$ string ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1 ~ = =$ null ? string.Empty : VAR1.ToString();   
+x+ string V0 = V1?.ToString() ?? string.Empty;
+
+# • Seed Edit
+
+${ \bf x } _ { - }$ Assert.True(Directory.Exists(V0) $\scriptstyle \mathbf { \mu = } \mathbf { \mu } \mathbf { V } 1$ );   
+$\mathbf { \pmb { x } } _ { + }$ Assert.Equal(Directory.Exists(V0), V1);
+
+# • Neighbor
+
+${ \bf x } _ { - }$ Assert.True(V0.GetString(V0.GetBytes(LITERAL)) $= =$ V1.ContainingAssembly.Identity.CultureName);   
+$\mathbf { \pmb { x } } _ { + }$ Assert.Equal(V0.GetString(VAR0.GetBytes(LITERAL)), V1.ContainingAssembly.Identity.CultureName);
+
+The seed and neighboring edits share functionally or syntactically similar patterns.
+
+# Examples
+
+The following edit is a related edit of the first example above, as it applies the same simplification (.Where(COND).Func() to .Func(COND)), but for FirstOrDefault instead of Single:
+
+• Seed Edit ${ \bf x } _ { - }$ var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ .Where( ${ \begin{array} { r l } { \ V { 2 } } & { { } = > } \end{array} }$ V2.Name $= =$ LITERAL).Single(); $\mathbf { \pmb { x } } _ { + }$ var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ .Single( ${ \bar { \mathsf { V } } } { \bar { \mathsf { Z } } } { = } { > }$ V2.Name $= =$ LITERAL);
+
+# • Neighbor
+
+${ \bf x } _ { - }$ var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ .Where( ${ \begin{array} { r l } { \ V { 2 } } & { { } = > } \end{array} }$ V3.ReportsTo $\scriptstyle \mathbf { \mu = } \mathbf { \mu } , \mathbf { \Lambda } _ { \mathsf { V 2 } }$ .EmployeeID).FirstOrDefault();   
+$\mathbf { \pmb { x } } _ { + }$ var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ .FirstOrDefault(V2 ${ } = > { }$ V3.ReportsTo $= =$ V2.EmployeeID);
+
+The following edit is a related edit of the second example above, as it also replaces a ternary expression for null checking with the ?. and ?? operators:
+
+• Seed Edit ${ \bf x } _ { - }$ Type ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ == null ? typeof(object) : V1.GetType(); $\mathbf { \pmb { x } } _ { + }$ Type $\begin{array} { r } { \nabla 0 \ \mathrm { ~  ~ } \nabla 1 \ ? } \end{array}$ .GetType() ?? typeof(object);
+
+# • Neighbor
+
+${ \bf x } _ { - }$ var ${ \mathrm { V } } 0 ~ = ~ { \mathrm { V } } 1$ != null ? V1.ToList() : new List<TextSpan $>$ ();   
+$\mathbf { \pmb { x } } _ { + }$ var $\begin{array} { r } { \nabla 0 \ \mathrm { ~  ~ } \nabla 1 \ ? } \end{array}$ .ToList() ?? new List<TextSpan>();
+
+We also considered pairs such as the following related, since they share similar syntactic structure
+
+• Seed Edit ${ \bf x } _ { - }$ V0.State = V1; $\mathbf { \pmb { x } } _ { + }$ V0.SetState(VAR1);
+
+• Neighbor ${ \pmb x } _ { - }$ V0.Quantity $\begin{array} { r l } { = } & { { } \nabla 1 } \end{array}$ ; x+ V0.SetQuantity(V1);
+
+# Rating 0 Not Related
+
+The seed and neighboring edits are not related based on the above criteria.
+
+Rating 2 Semantically and Syntactically Equivalent
+
+The changed constituents in the seed edit and the neighboring edit are applied to the similar positions of the original sentence, serving the same syntactic and semantic role. For example,   
+
+<table><tr><td>Seed Edit</td><td>Neighbor</td></tr><tr><td>chaz guest（born 196l ） was born in ni- agra falls，...,a decorated hero in wwii in europe ,including the purple heart .</td><td>randal l. schwartz（ born november 22, →1961),also known as merlyn,is an amer- ican author，system administrator and pro- gramming consultant.</td></tr><tr><td>he was elected to donegal county council for sinn fin in 1979 ,and held his seat until his death  at age 56.</td><td>davis graduated from high school in january 1947,immediately enrolling at wittenberg college in rural ohio at age 17.</td></tr><tr><td>●drorfeiler served as a paratrooper in the israel defense forces.</td><td>●nagaurfort - sandy fort ; centrally located ； 2nd century old ；witnessed many battles ; lofty walls &amp; spacious campus ; having many palaces &amp; temples inside .</td></tr><tr><td>the original old bay house,home of the chief factor,still exists and is now part of the fort vermilion national historic site.</td><td>the population was 6,400 at the 2010 cen- sus and ispartof the st.louismetropolitan area.</td></tr></table>
+
+Rating 1 Syntactically Related
+
+The changed constituents in the seed and the neighboring edit are applied to the similar positions of the original sentence, and they play similar syntactic roles. This includes examples like adding a disfunction, adding a complement, prepositional clause or other syntactic constructs with similar phrases or language structures. For example,   
+
+<table><tr><td>Seed Edit</td><td>Neighbor</td></tr><tr><td>the douro fully enters portuguese territory just after the confluence with the gueda river ; once the douro enters portugal,major pop- ulation centres are less frequent?along the river.</td><td>she made a brief return to the screen in ” par- rish ”（1961）,playing the supporting role of mother which received little attention  by the press.</td></tr><tr><td>of pagumon living there instead who imme- diately proceeded to treat the digidestined as honored guests ，saying that pagumon are the fresh form of koromon</td><td>when they found it,they discovered a groupin 2O12 slote and his baseball book ” jake ” were the subject of an espn（30 for 30） short documentary in which slote describes his writing process and reads from the book D,saying it is his best writing.</td></tr><tr><td>and supplied as a complete ready - to - fly - aircraft for the flight training and aerial work markets .</td><td>the aircraft was intended to be certifiedin june reinforcements finally did arrive when ?provincial andmilitia units from new york,new jersey,and new hampshire were sent up from fort edward by general daniel webb .</td></tr></table>
+
+<table><tr><td>Rating 0 Not Related</td></tr><tr><td>The seed and neighboring edits are not related based on the above criteria.</td></tr></table>

parse/train/BkeOp6EKDH/BkeOp6EKDH.md

@@ -0,0 +1,202 @@
+# TRIMAP: LARGE-SCALE DIMENSIONALITY REDUCTION USING TRIPLETS
+
+# ABSTRACT
+
+We introduce “TriMap”; a dimensionality reduction technique based on triplet constraints that preserves the global accuracy of the data better than the other commonly used methods such as t-SNE, LargeVis, and UMAP. To quantify the global accuracy, we introduce a score which roughly reflects the relative placement of the clusters rather than the individual points. We empirically show the excellent performance of TriMap on a large variety of datasets in terms of the quality of the embedding as well as the runtime. On our performance benchmarks, TriMap easily scales to millions of points without depleting the memory and clearly outperforms t-SNE, LargeVis, and UMAP in terms of runtime.
+
+# 1 INTRODUCTION
+
+Data visualization based on dimensionality reduction (DR) is a core problem in data analysis and machine learning. The aim of DR is to provide a low-dimensional representation (typically in 2D or 3D) of a given high-dimensional dataset that preserves the overall structure of the data as much as possible. The earlier approaches for DR involve linear methods such as PCA $( | \mathbf { P e a r s o n } | , | 1 9 0 1 | )$ . PCA aims to maintain the second-order statistics of the data by projecting the points into the low dimensional space that preserves the maximum amount of variance among all such projections. As a result, PCA has been shown to be effective in preserving the global structure of the data $\mathrm { ( | \overline { { { S i l v a } } } } |$ & Tenenbaum, 2003). The global structure includes the overall shape of the dataset, placement of the clusters, and existence of potential outliers. Unlike PCA, much of the focus of the more recent non-linear methods including t-SNE (Maaten & Hinton, $\textcircled { 2 0 0 8 }$ , LargeVis $\left( \mathbb { T a n g e t a l . } \right) , \left[ 2 0 1 6 \right)$ and UMAP (McInnes et al., 2018) has been on preserving the local neighborhood structure of each individual point. Similarly, the common performance measures of DR such as trustworthinesscontinuity (Venna & Kaski, 2005), precision-recall (i.e. AUC) $\left( \mathrm { N e n n a ~ e t ~ a l . } \right) \left[ \mathrm { 2 0 1 0 } \right]$ , and nearestneighbor accuracy have also been developed by retaining the same focus on reflecting the local accuracy of the embedding. Thus, there has been a lack of attention on developing methods that focus on preserving the global structure of the data and likewise, practical performance measures to assess the global accuracy.
+
+We first introduce the global score, a quantitative measure which reflects the closeness of a given embedding to the PCA embedding (which is optimal by means of preserving the data variance). The purpose of this score is to measure the accuracy of an embedding in reflecting the overall placement of the clusters of points relative to their original representation in high-dimension. By design, PCA yields the highest global score among all the DR methods and high values of global score indicates the efficacy of a DR method in reflecting the global structure.
+
+Next, we introduce TriMap, a DR method which focuses on preserving the global structure of the data in the embedding. Pairwise (dis)similarities between points (used by the previous DR methods) seem to be insufficient in capturing the global structure. Instead, TriMap incorporates a higher order of structure to construct the embedding by means of triplets:
+
+$$
+( i , j , k ) \Leftrightarrow p o i n t i s c l o s e r t o p o i n t j t h a n p o i n t k .
+$$
+
+The key idea behind TriMap stems from semi-supervised metric learning $\left( \overline { { \mathrm { A m i d e t a l . } } } , \overline { { 2 0 1 6 } } \right)$ : Given an initial low-dimensional representation for the data points, the triplet information from the highdimensional representation of the points is used to enhance the quality of the embedding. Similarly, TriMap is initialized with the low dimensional PCA embedding, and this embedding is then modified using a set of carefully selected triplets from the high-dimensional representation.
+
+![](images/126550d3d027ba058dc4c638816e25f31438dc6863e0896965d53054c49a4bed.jpg)  
+Figure 1: 2-D Visualizations of the S-curve dataset: (a) original dataset in 3-D, (b) t-SNE, (c) UMAP, (d) TriMap, and (e) PCA. The values of AUC and global score, for respectively measuring local and global accuracy, are shown in order as a pair (AUC, GS) for each embedding. Despite having higher AUC values, t-SNE and UMAP both fail to reflect the overall shape of the S-curve. On the other hand, TriMap successfully unveils the underlying structure in the original dataset. Note that GS is the only DR performance measure that can reflect this property.
+
+With an extensive set of experiments, we show that TriMap produces excellent results on a variety of real-world as well as synthetic datasets. We show that in many cases TriMap outperforms all the competitor non-linear methods by means of global score and provides comparable local accuracy. While being significantly faster than t-SNE, TriMap provides comparable runtime to UMAP and LargeVis while scaling drastically better to larger datasets. On the Character Font Images dataset of ${ \sim } 1 . 7 \mathbf { M }$ points, TriMap calculates the embedding in ${ \sim } 1 . 3$ hours while LargeVis takes more than 3 hours and UMAP exceeds the 12 hours time limit. Our contributions can be summarized as follows:
+
+• We introduce a global score to quantify the quality of a low-dimensional embedding in reflecting the global structure of the high-dimensional data such as placement of the clusters rather than the local neighborhood of individual points.   
+We introduce TriMap, a fast dimensionality reduction method which provides embeddings of the data that are globally more accurate than other non-linear DR methods such as t-SNE, LargeVis, and UMAP.   
+• We provide an efficient implementation1 of TriMap that can easily scale to millions of points on commodity hardware and outperforms the competing methods in terms of runtime. We also perform many large-scale experiments on various datasets to show the efficacy of TriMap in terms of DR performance measures and runtime.
+
+# 2 A MEASURE OF GLOBAL ACCURACY
+
+Consider the S-curve dataset2 which consists of 5000 points in 3-D uniformly sampled from an S-shaped manifold (Figure 1.(a)). This dataset serves as a paradigmatic problem for evaluating the performance of DR methods. In Figure $\bigstar \bigstar$ we show the results of 2-D embeddings of the S-curve dataset using t-SNE, UMAP, TriMap, and PCA. The top of each graph is labeled by the scoring pair (AUC, GS) where GS stands for global score (introduced below). Note that both t-SNE and UMAP provide higher values of the AUC score and locally preserve the continuity of the manifold. However, they both fail to recover the global structure of the S-curve, which is naturally reflected in the PCA embedding. On the other hand, our TriMap method (formally defined later) successfully recovers the structure of the S-curve by “unveiling” the curved shape of the manifold at both ends. Overall, the 2-D TriMap embedding resembles the original 3-D representation as much as possible. Note also that GS is the only measure that can reflect the global accuracy of the embedding.
+
+The previous example indicates that the local measures of DR performance (such as AUC) cannot reflect the global accuracy of a low-dimensional embedding. In fact, the low-granular structure of the data can only be estimated by considering the global statistics of the dataset, as regarded by the PCA method. PCA is a linear DR method that projects the high-dimensional data onto the top- $d$ orthogonal directions having the highest variance. In order to calculate the mapping, PCA only considers the aggregate statistics of the dataset rather than the local information of each individual data point. As a result, PCA is extremely well suited at retaining the global structure of the data, i.e. the overall shape of the dataset, placement of the clusters, and existence of potential outliers.
+
+![](images/90058bdad23fe495c6d48369d44b01219a795fa12a3178643624027d56f79383.jpg)  
+Figure 2: The Effect of the weight transformation on the MNIST dataset: (a) no weight transformation, (b) $\gamma = 5 0$ , (c) $\gamma = 5 0 0$ (default), and (d) $\gamma = 5 0 0 0$ . The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure. Larger values of $\gamma$ emphasizes more on the local accuracy rather than the global accuracy.
+
+However, by focusing on the global structure, PCA loses much of the local information such as the neighborhood structure of each data point.
+
+Given a low-dimensional mapping produced by PCA, it is possible to calculate an optimal inverse mapping into the original high-dimensional space by means of minimizing the squared error. The optimal inverse map also corresponds to a linear mapping3. In order to quantify the global accuracy of a DR result, we focus on the accuracy of the embedding in reflecting the global structure of the data similar to PCA. That is, we consider the minimum reconstruction error of the original dataset by means of a linear inverse map. Given $n$ data points $\{ \pmb { x } _ { i } \in \mathbb { R } ^ { m } \} _ { i = 1 } ^ { n }$ , let $\pmb { X } \in \mathbb { R } ^ { m \times \bar { n } }$ denote the high-dimensional data matrix where the $i$ -th column corresponds to $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ . Similarly, let $\pmb { Y } \in \mathbb { R } ^ { d \times n }$ denote the matrix of the low-dimensional embedding of the points $\{ \pmb { y } _ { i } \in \mathbb { R } ^ { d } \} _ { i = 1 } ^ { n }$ . Without loss of generality, we assume both $\boldsymbol { X }$ and $\mathbf { Y }$ are centered. We define the Minimum Reconstruction Error $( M R E )$ from the embedding as
+
+$$
+\mathcal { E } ( \boldsymbol { Y } | \boldsymbol { X } ) : = \operatorname* { m i n } _ { \boldsymbol { A } \in \mathbb { R } ^ { m \times d } } \| \boldsymbol { X } - \boldsymbol { A } \boldsymbol { Y } \| _ { \mathrm { F } } ^ { 2 } ,
+$$
+
+where $\| \cdot \| _ { \mathrm { F } }$ denotes the Frobenius norm4. Note that PCA has the lowest possible MRE among all the DR methods. Thus, in order to obtain a normalized measure of global accuracy of a given embedding $\mathbf { Y }$ for a data $\boldsymbol { X }$ , we define the global score $( G S )$ as
+
+$$
+\mathrm { G S } ( Y | X ) : = \exp \Big ( - \frac { \mathcal { E } ( Y | X ) - \mathcal { E } _ { \mathrm { P C A } } } { \mathcal { E } _ { \mathrm { P C A } } } \Big ) \in [ 0 , 1 ] ,
+$$
+
+where $\mathcal { E } _ { \mathrm { P C A } } : = \mathcal { E } ( Y _ { \mathrm { P C A } } | X )$ denotes the MRE achieved by the PCA embedding $\mathbf { Y } _ { \mathrm { P C A } }$ on the same dataset $\boldsymbol { X }$ . Note that $\mathrm { \bf G S } ( Y _ { \mathrm { P C A } } | X ) = 1$ and we claim that larger values of GS indicate a higher capacity of a DR method to reflect the global structure of the data, as shown in the experiments.
+
+In the remainder of the paper, we use GS as the global measure of performance. Due to the high computational complexity for calculating the trustworthiness-continuity and AUC scores for large data sets, we use nearest-neighbors accuracy as the local measure of performance henceforth.
+
+# 3 THE TRIMAP METHOD
+
+We now formally introduce the TriMap method. Recall that a triplet consists of three points $( i , j , k )$ where point $i$ is closer to point $j$ than point $k$ . TriMap chooses a subset $\mathcal { T } = \{ ( i , j , \bar { k } ) \}$ of triplets and assigns a weight $\omega _ { i j k } \geq 0$ for each triplet: a higher value of $\omega _ { i j k }$ implies that the pair $( i , k )$ is located much farther than the pair $( i , j )$ . We define the loss of the triplet $( i , j , k )$ as
+
+$$
+\ell _ { i j k } : = \omega _ { i j k } \frac { s ( y _ { i } , y _ { k } ) } { s ( y _ { i } , y _ { j } ) + s ( y _ { i } , y _ { k } ) } , \mathrm { ~ w h e r e ~ } s ( y _ { i } , y _ { j } ) = \left( 1 + | y _ { i } - y _ { j } | | ^ { 2 } \right) ^ { - 1 } ,
+$$
+
+![](images/922bb546e53ad0604be26eae945edea542602a5fe252f6a85aabb42a6bfac050.jpg)  
+Figure 3: Effect of changing the number of triplets on the quality of the embeddings of the MNIST dataset. We consider $( m , m ^ { \prime } , r ) = c \times ( 2 , 1 , 1 )$ for: (a) $c = 1$ , (b) $c = 2$ , (c) $c = 5$ (default), (d) $c = 1 0$ , and (e) $c = 2 0$ . The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure. The quality of embedding does not improve significantly after adding a certain number of triplets.
+
+is a similarity function between $\mathbf { \nabla } _ { \mathbf { \boldsymbol { y } } _ { i } }$ and ${ \pmb y } _ { j }$ . The choice of $s$ is motivated by the good performance of Student t-distribution for similarities in low-dimension in the t-SNE method. Note that the loss of the triplet $( i , j , k )$ approaches zero as $\| \pmb { y } _ { i } - \pmb { y } _ { j } \|$ decreases and $\| \pmb { y } _ { i } - \pmb { y } _ { k } \|$ increases.
+
+We first develop the weighing scheme for the triplets. To reflect the relative similarities in high-dimension, we define the unnormalized weight of the triplet $( i , j , k )$ as
+
+$$
+\tilde { \omega } _ { i j k } = \exp ( d _ { i k } ^ { 2 } - d _ { i j } ^ { 2 } ) \geq 0 ,
+$$
+
+in which, $d _ { i j }$ is any distance measure between $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ and $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { j } }$ in highdimension. For Euclidean distances, we use the scaling introduced in (Zelnik-Manor & Perona, $\textcircled { 2 0 0 5 }$ ,
+
+$$
+d _ { i j } ^ { 2 } = \frac { \lVert \pmb { x } _ { i } - \pmb { x } _ { j } \rVert ^ { 2 } } { \sigma _ { i j } } ,
+$$
+
+where $\sigma _ { i j } = \sigma _ { i } \sigma _ { j }$ and $\sigma _ { i }$ is set to the average Euclidean distance between $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ and the set of nearest-neighbors of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ from 4-th to 6-th neighbors. This choice of $\sigma _ { i j }$ adaptively adjusts the scaling based on the density of the data.
+
+![](images/ccbf9f3fa2e2973959a501e01799674558f18b1188314acb65e1070afdf301e8.jpg)  
+Figure 4: $\gamma$ -scaled logtransformation with different values of $\gamma$ . The value NIL corresponds to no transformation.
+
+While the choice of weights $\tilde { \omega } _ { i j k }$ works well in practice, we adjust the weights further by applying a non-linear transformation that emphasizes the smaller weights. Expanding the values of small weights has the effect of placing the nearest-neighbors closer to the point and pushing the remaining points farther away, thus improving the local accuracy (as shown in Figure $\bigtriangledown$ and discussed later). The final value of the weight $\omega _ { i j k }$ is obtained by applying the $\gamma$ -scaled log-transformation (see Figure 4),
+
+$$
+\omega _ { i j k } = \zeta _ { \gamma } \big ( \frac { \tilde { \omega } _ { i j k } } { \gamma } + \delta \big ) \quad \mathrm { w h e r e } \zeta _ { \gamma } ( u ) : = \log \big ( 1 + \gamma u \big ) ,
+$$
+
+in which $\mathcal { W } = \mathrm { m a x } _ { ( i ^ { \prime } , j ^ { \prime } , k ^ { \prime } ) \in \mathcal { T } } \tilde { \omega } _ { i ^ { \prime } j ^ { \prime } k ^ { \prime } } , ~ \gamma > 0$ is a scaling factor, and $\delta$ is a small constant. We use $\gamma = 5 0 0$ and $\delta = 1 0 ^ { - 4 }$ in all our experiments.
+
+To construct the embedding, we consider a small subset of all possible triplets $( i , j , k )$ for which, the closer point $j$ belongs to the set of nearest-neighbors of the point $i$ and the farther point $k$ is among the points that are more distant from $i$ than $j$ , chosen uniformly at random. For each point we consider its $m = 1 0$ nearest neighbors and sample $m ^ { \prime } = 5$ triplets per nearest-neighbor. This yields $m \times m ^ { \prime } = 5 0$ nearest-neighbor triplets per point. In addition, we also add $r = 5$ random triplets $( i , j , k )$ per each point $i$ where $j$ and $k$ are sampled uniformly at random and their order is possibly switched based on their nearness to $i$ . This yields $m \times m ^ { \prime } + r = 5 5$ triplets per point in total. Thus, the overall complexity of the optimization step is linear in number of points $n$ . The computational complexity is dominated by the nearest-neighbor search, which is shared among all the recent methods such as t-SNE, LargeVis, and UMAP. We use ANNOY for the approximate nearest-neighbor search5 which is based on random projection trees.
+
+While a random initialization for the embedding also works well in practice, we initialize the embedding to the PCA solution $\mathbf { Y } _ { \mathrm { P C A } }$ (scaled by a small constant value for better convergence). The PCA initialization for TriMap allows faster convergence while preserving much of the global structure discovered by PCA. Note that the other DR methods such as t-SNE are extremely sensitive to the initialization and do not converge well with any initial solution other than small random initialization around the origin.
+
+Table 1: Runtime of the methods in hh:mm:ss format on single machine with $2 . 6 \ : \mathrm { G H z }$ Intel Core i5 CPU and 16 GB of memory. We limit the runtime of each method to 12 hours. Also, UMAP runs out of memory on datasets larger than ${ \sim } 4 \mathbf { M }$ points.   
+
+<table><tr><td>Dataset (size)</td><td>t-SNE</td><td>LargeVis</td><td>UMAP</td><td>TriMap</td><td>Speedup</td></tr><tr><td>COIL-20 (1440)</td><td>00:00:08</td><td>00:05:51</td><td>00:00:04</td><td>00:00:02</td><td>2.00×</td></tr><tr><td>USPS (11K)</td><td>00:02:02</td><td>00:06:12</td><td>00:00:12</td><td>00:00:11</td><td>1.10×</td></tr><tr><td>Epileptic Seizure (11.5K)</td><td>00:03:11</td><td>00:06:17</td><td>00:00:15</td><td>00:00:12</td><td>1.25×</td></tr><tr><td>20 Newsgroup (18K)</td><td>00:05:34</td><td>00:06:57</td><td>00:00:26</td><td>00:00:21</td><td>1.24×</td></tr><tr><td>TabulaMuris (54K)</td><td>00:17:32</td><td>00:09:29</td><td>00:01:12</td><td>00:01:06</td><td>2.00×</td></tr><tr><td>MNIST(70K)</td><td>00:20:38</td><td>00:11:29</td><td>00:01:15</td><td>00:01:23</td><td>0.90×</td></tr><tr><td>FashionMNIST(70K)</td><td>00:19:10</td><td>00:11:04</td><td>00:01:18</td><td>00:01:24</td><td>0.93×</td></tr><tr><td>TVNews (~129K)</td><td>00:38:59</td><td>00:16:26</td><td>00:02:57</td><td>00:02:45</td><td>1.07×</td></tr><tr><td>360+KLyrics (~360K)</td><td>08:50:49</td><td>00:44:16</td><td>00:25:23</td><td>00:13:49</td><td>1.84×</td></tr><tr><td>Covertype(~581K)</td><td>二</td><td>00:44:54</td><td>02:59:41</td><td>00:24:42</td><td>1.82×</td></tr><tr><td>RCV1 (800K)</td><td>二</td><td>01:34:38</td><td>04:55:53</td><td>00:36:59</td><td>2.56×</td></tr><tr><td>Character Font Images (~1.7M)</td><td>二</td><td>03:16:19</td><td>二</td><td>01:17:50</td><td>2.52×</td></tr><tr><td>KDDCup99 (~4.9M)</td><td>二</td><td>二</td><td>二</td><td>04:17:01</td><td>二</td></tr><tr><td>HIGGS (11M)</td><td>1</td><td>1</td><td>1</td><td>10:08:36</td><td>1</td></tr></table>
+
+We define the final loss as the sum of the losses of the sampled triplets in $\tau$
+
+$$
+\ell _ { \mathrm { T r i M a p } } = \sum _ { ( i , j , k ) \in \mathcal { T } } \ell _ { i j k } .
+$$
+
+The loss is minimized using the full-batch gradient descent with momentum using the delta-bar-delta method. In all our experiments, we perform 400 iterations with the value of momentum parameter equal to 0.5 during the first 250 iterations and 0.8 afterwards.
+
+Finally, note that there exists connections between TriMap and a number of triplet (aka ordinal) embedding methods such as t-STE (Van Der Maaten & Weinberger, $\boxed { 2 0 1 2 } )$ . The triplet embedding methods have been developed for a different setting where the goal is to find an embedding based on a given pre-specified set of triplets obtained from human evaluators (or some form of implicit feedback). For instance, t-STE maximizes the sum of log of the satisfaction probabilities of the triplets to calculate the embedding. It is worth mentioning that TriMap is a DR method that is designed to sample the informative triplets from the high-dimensional representation of a set of points and assign weights to these triplets to reflect the relative similarities of these points. Although TriMap can also be used for the triplet embedding task, we only focus on the DR results 6.
+
+# 3.1 EFFECT OF DIFFERENT PARAMETERS
+
+We briefly discuss the effect of different parameters, namely the total number of triplets $| \tau |$ and the $\gamma$ -scaled log-transformation, on the quality of the embedding. TriMap is particularly robust to the number of sampled triplet for constructing the embedding. This can be explained by the high amount of redundancy in the triplets (the triplets $( i , j , k )$ and $\bar { ( } i , j , k ^ { \prime } )$ convey the same information if $k$ and $k ^ { \prime }$ are nearest neighbors and also mapped nearby). In Figure $\textcircled { 3 }$ we consider various values for $m$ , $m ^ { \prime }$ , and $r$ for the MNIST dataset while fixing the remaining parameters. In fact, using large number of triplets can sometimes introduce an overhead and require larger number of iterations to converge.
+
+A more important parameter is $\gamma$ which controls the trade-off between the local and global accuracy. Larger values of $\gamma$ increases the relative importance of triplets with smaller weights. This causes the method to focus on the nearest-neighbor points rather than the points that are far away, thus improving the local accuracy. On the other hand, improving the local accuracy can impair the global accuracy. In Figure $^ { 2 , }$ we plot the $\gamma$ -scaled log-transformation for various $\gamma$ values and illustrate the results on MNIST without the log-transformation as well as the results with different $\gamma$ values. For larger values of $\gamma$ the clusters tend to become more compressed and as a result, the nearest-neighbor accuracy is improved. On the other hand, the global score starts to decrease for larger $\gamma$ values.
+
+![](images/2ce1b1fc67b1b7febaf375f4781a2a98d8cfd034c15252995fbc93981b425354.jpg)  
+Figure 5: Visualizations of different datasets using t-SNE, UMAP, TriMap, and PCA. Each row corresponds to one dataset and each column represents one method. The values of nearest neighbor accuracy and global score are shown as a pair (NN,GS) on top of each figure.
+
+# 4 EXPERIMENTS
+
+In this section, we apply TriMap on a set of real-world as well as synthetic datasets and compare the results to t-SNE, LargeVis, UMAP, and PCA methods. The datasets used in our experiments are listed in Table 1 and a short description is given in the appendix. All experiments are conducted on a single machine with $2 . 6 \ : \mathrm { G H z }$ Intel Core i5 CPU and $1 6 \mathrm { \ G B }$ of memory. We limit the runtime of each algorithm to 12 hours. For implementations, we use the default sklearn implementation for t-SNE and the official implementations of LargeVis and UMAP provided by the authors7,8. Due to lack of space, we provide the comparison to the LargeVis results as well additional TriMap results on the larger datasets in the appendix.
+
+![](images/c30bd6f21a0a26542831274847c6c7fa12491842072df61b0d6c9e9ef3c98985.jpg)  
+Figure 5: Visualizations of different datasets (continued) using t-SNE, UMAP, TriMap, and PCA. Each row corresponds to one dataset and each column represents one method. The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure.
+
+In order to have a fair comparison, we use the default parameter values for all methods, including ours $\mathbf { \Phi } _ { m } = 1 0 \mathbf { \Phi } $ , $m ^ { \prime } = 5$ , $r \ = \ 5$ , $\gamma ~ = ~ 5 0 0$ , and 400 iterations). Also to reduce the overhead induced by the dimensionality of the data in the nearest-neighbor search step, we reduce the number of dimensions of the dataset to 100 if necessary, using the PCA method. To evaluate the local performance, we show the nearest-neighbor accuracy of each result. We also show the GS as a measure of global performance. The performance measures are shown on top of each figure as a pair (NN, GS).
+
+# 4.1 RUNTIME
+
+The runtime of the methods are provided in Table $^ 1$ in the hh:mm:ss format. We limit the runtime of each method to 12 hours. As can be seen from the results, TriMap provides excellent runtime and outperforms all the other methods in most cases. Also, TriMap easily scales to millions of points while the other methods exceed the time limit or run out of memory. For instance, UMAP causes an out of memory error for datasets larger than ${ \sim } 4 \mathbf { M }$ points.
+
+# 4.2 VISUALIZATIONS
+
+The visualizations of the datasets using TriMap as well as the other competing methods are shown in Figure $5$ and $\bigtriangledown$ For some results, we provide a zoomed in snippet over the main figure to provide a more detailed illustration. Overall, TriMap preserves the underlying global structure of the data better than the other competing methods. This is reflected by the larger GS values for TriMap as well as visually comparing the embeddings to the PCA result. For example, TriMap recovers the continuous structure of the TV news dataset and separates the remaining outliers in the data which are also identified by the PCA method. This can be verified by comparing the placement of an example outlier point, marked with a red $\times$ , by the different methods: TriMap shows this point among other outliers whereas t-SNE and UMAP fail to uncover this information. Also, the global score of TriMap on this dataset is much higher than the other methods. Further discussion is given in the appendix.
+
+![](images/96f7feaeec77ebd9f9917f45d0ce034c8fd693edd822483e4711782fd49320fc.jpg)  
+Figure 6: Visualizations of Covertype and RCV1 datasets using UMAP, TriMap, and PCA, and visualizations of the Character Font Images dataset using LargeVis, TriMap, and PCA. The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure.
+
+# 5 CONCLUSION AND FUTURE WORK
+
+TriMap is a fast and efficient method that can be easily applied to large datasets. While TriMap is extremely effective for uncovering the global structure of the data, other methods such as t-SNE can provide additional insight about the local neighborhood of individual points. As a future research direction, we consider using pairwise constraints along with triplet constraints to improve the local accuracy. The current implementation of TriMap utilizes a single core. Parallel implementation of the method that can exploit multiple cores is another future direction. Furthermore, the global accuracy is measured in terms of the global score which is based on the assumption that linear projection obtained by PCA is globally optimal. While our global score can provide insight about the global accuracy of the embedding in many cases, it appears to be ineffective when the data is highly non-linear or contains a large amount of outliers. Developing non-linear and more robust global performance measures could significantly improve the assessment of the DR results and provide guidelines for developing more accurate DR techniques.
+
+# REFERENCES
+
+Ehsan Amid, Aristides Gionis, and Antti Ukkonen. Semi-supervised kernel metric learning using relative comparisons. arXiv preprint arXiv:1612.00086, 2016.   
+Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(Nov):2579–2605, 2008.   
+Leland McInnes, John Healy, and James Melville. Umap: Uniform manifold approximation and projection for dimension reduction. arXiv preprint arXiv:1802.03426, 2018.   
+Karl Pearson. Liii. on lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11):559–572, 1901.   
+Vin D Silva and Joshua B Tenenbaum. Global versus local methods in nonlinear dimensionality reduction. In Advances in neural information processing systems, pp. 721–728, 2003.   
+Jian Tang, Jingzhou Liu, Ming Zhang, and Qiaozhu Mei. Visualizing large-scale and highdimensional data. In Proceedings of the 25th international conference on world wide web, pp. 287–297. International World Wide Web Conferences Steering Committee, 2016.   
+Laurens Van Der Maaten and Kilian Weinberger. Stochastic triplet embedding. In 2012 IEEE International Workshop on Machine Learning for Signal Processing, pp. 1–6. IEEE, 2012.   
+Jarkko Venna and Samuel Kaski. Local multidimensional scaling with controlled tradeoff between trustworthiness and continuity. In Proceedings of 5th Workshop on Self-Organizing Maps, pp. 695–702. Citeseer, 2005.   
+Jarkko Venna, Jaakko Peltonen, Kristian Nybo, Helena Aidos, and Samuel Kaski. Information retrieval perspective to nonlinear dimensionality reduction for data visualization. Journal of Machine Learning Research, 11(Feb):451–490, 2010.   
+Lihi Zelnik-Manor and Pietro Perona. Self-tuning spectral clustering. In Advances in neural information processing systems, pp. 1601–1608, 2005.
+
+# A DATASETS
+
+The datasets used in the experiments are listed below. All datasets are publicly available online and a download link is provided.
+
+• COIL- ${ \bf \nabla } \cdot 2 { \bf 0 } ^ { 9 }$ (1440): gray-scale images of 20 objects in uniformly sampled orientations (5 degrees of rotation, 72 images per object). Each image is pre-processed by having the background removed and cropped into size $1 2 8 \times 1 2 8$ . $\mathbf { U S P S } ^ { 1 0 }$ (11K): images of handwritten digits (0–9) of size $1 6 \times 1 6$ . Epileptic Seizure11 (11.5K): EEG signal recordings of brain activity for seizure recognition. It contains 178-dimensional vectors belonging to 5 categories.   
+20 Newsgroup11 (18K): newsgroup posts categorized into 20 topics. We use a TF-IDF representation of the words in each document as the features.   
+Tabula Muris12 $( \sim 5 4 \mathrm { K } )$ : single cell transcriptome data from the mouse from 20 organs.   
+• MNIST13 (70K): images of handwritten digits (0–9) of size $2 8 \times 2 8$ .   
+• Fashion MNIST14 (70K): gray-scale images of clothing items such as t-shirt, pullover, bag, etc. of size $2 8 \times 2 8$ .   
+TV News11 $( \sim 1 2 9 \mathrm { K } )$ : audio-visual features from TV news broadcast categorized into commercial and non-commercial.   
+• $\mathbf { 3 6 0 K + }$ Lyrics15 $( \sim 3 6 2 \mathrm { K } )$ : lyrics of songs from 12 different genres. We group similar genres together (metal-rock, R&B-pop, etc.) to form 7 groups. We use the TF-IDF representation of the words in the song as the features. Covertype11 $( \sim 5 8 1 \mathrm { K } )$ : cartographic features for forest cover type prediction.   
+• $\mathbf { R C V 1 } ^ { 1 6 }$ (800K): Reuters Corpus Volume I archive of categorized newswire stories. Character Font Images11 $( { \sim } 1 . 7 \mathrm { M } )$ : images of character from scanned and computer generated fonts.   
+• $\mathbf { K D D C u p 9 9 } ^ { \mathrm { 1 1 } }$ $( { \sim } 4 . 9 \mathrm { { M } ) }$ : computer network intrusion detection.   
+• $\mathbf { H } \mathbf { I } \mathbf { G } \mathbf { G } \mathbf { S } ^ { 1 1 }$ (11M): Higgs bosons recognition from a background process.
+
+# B MORE VISUALIZATIONS
+
+We compare the results of TriMap to LargeVis in Figure 7 and 8. We also provide more visualizations obtained using TriMap in Figure 9.
+
+# C DISCUSSION
+
+We briefly discuss the results of TriMap and draw a comparison to the other methods.
+
+TriMap generally provides better global accuracy compared to the competing methods. It also successfully maintains the continuity of the underlying manifold. This can be seen from the COIL-20 result where certain clusters are located farther away from the remaining clusters. However, the underlying structure for the main cluster resembles the one provided by the other methods. TriMap also preserves the continuous structure in the Fashion MNIST and the TV News datasets.
+
+TriMap is also efficient in uncovering the possible outliers in the data. For instance, PCA reveals a large number of outliers in the Tabula Muris and the $3 6 0 \mathrm { + K }$ Lyrics datasets. These outliers are located far away from the main clusters in the TriMap results. However, the same points are located very close to the remaining points in the t-SNE results.
+
+<table><tr><td>http://www.cs.columbia.edu/CAvE/software/softlib/coil-20.php</td></tr><tr><td>https://www.kaggle.com/bistaumanga/usps-dataset</td></tr><tr><td>http://archive.ics.uci.edu/ml/index.php</td></tr><tr><td>https://tabula-muris.ds.czbiohub.org/</td></tr><tr><td>http://yann.lecun.com/exdb/mnist/</td></tr><tr><td>https://github.com/zalandoresearch/fashion-mnist</td></tr><tr><td>https://www.kaggle.com/gyani95/380000-lyrics-from-metrolyrics</td></tr><tr><td>https://scikit-learn.org/0.18/datasets/rcv1.html</td></tr></table>
+
+![](images/1b3d08ee2fd1e362cbb60fdc07451f5e9efbec7336f6557e5cc45d24a6ff6924.jpg)  
+Figure 7: Visualizations of different datasets using LargeVis, TriMap, and PCA. Each row corresponds to one dataset and each column represents one method. The values of nearest neighbor accuracy and global score are shown as a pair (NN,GS) on top of each figure.
+
+![](images/02894110afa287f8baca3934fc4118784cc7dec51a228d2a276abdd1c244e9a6.jpg)  
+Figure 7: Visualizations of different datasets (continued) using LargeVis, TriMap, and PCA. Each row corresponds to one dataset and each column represents one method. The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure.
+
+Additionally, both t-SNE and LargeVis tend to form spurious clusters by splitting the underlying connected manifold. This can be seen from the TV News results and the result of LargeVis on the Covertype dataset.
+
+Finally, notice that in some cases GS fails to reflect the global accuracy of the embeddings. This can be seen from the low GS values for all methods on the Covertype dataset. GS may become uninformative when there exists a high degree of non-linearity in the data that cannot be reflected using PCA. GS also cannot reflect the accuracy of the embedding in uncovering single outliers. Developing more accurate global measures for these scenarios is a future research direction.
+
+![](images/cf9669f5038690e85b2e3c5971d63271a86fb8c99aa4da0cb5ee4f8f808801dd.jpg)  
+Figure 8: Visualizations of Covertype and RCV1 datasets using LargeVis, TriMap, and PCA, and visualizations of the Character Font Images dataset using LargeVis, TriMap, and PCA. The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure.
+
+![](images/3861d063bab8a11cd7e97eb87ff420798a417be26fec70be2875a5b800d7374e.jpg)  
+Figure 9: Visualizations of KKDCup99 and HIGGS datasets TriMap and PCA. Each row corresponds to one dataset and each column represents one method. The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure. TriMap shows more structure for both datasets than PCA. Note that GS is uninformative for the KDDCup99 dataset.

parse/train/BkeOp6EKDH/BkeOp6EKDH_content_list.json

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+    {
+        "type": "text",
+        "text": "TRIMAP: LARGE-SCALE DIMENSIONALITY REDUCTION USING TRIPLETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            691,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            220,
+            544,
+            236
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We introduce “TriMap”; a dimensionality reduction technique based on triplet constraints that preserves the global accuracy of the data better than the other commonly used methods such as t-SNE, LargeVis, and UMAP. To quantify the global accuracy, we introduce a score which roughly reflects the relative placement of the clusters rather than the individual points. We empirically show the excellent performance of TriMap on a large variety of datasets in terms of the quality of the embedding as well as the runtime. On our performance benchmarks, TriMap easily scales to millions of points without depleting the memory and clearly outperforms t-SNE, LargeVis, and UMAP in terms of runtime. ",
+        "bbox": [
+            233,
+            250,
+            766,
+            376
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            398,
+            336,
+            415
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Data visualization based on dimensionality reduction (DR) is a core problem in data analysis and machine learning. The aim of DR is to provide a low-dimensional representation (typically in 2D or 3D) of a given high-dimensional dataset that preserves the overall structure of the data as much as possible. The earlier approaches for DR involve linear methods such as PCA $( | \\mathbf { P e a r s o n } | , | 1 9 0 1 | )$ . PCA aims to maintain the second-order statistics of the data by projecting the points into the low dimensional space that preserves the maximum amount of variance among all such projections. As a result, PCA has been shown to be effective in preserving the global structure of the data $\\mathrm { ( | \\overline { { { S i l v a } } } } |$ & Tenenbaum, 2003). The global structure includes the overall shape of the dataset, placement of the clusters, and existence of potential outliers. Unlike PCA, much of the focus of the more recent non-linear methods including t-SNE (Maaten & Hinton, $\\textcircled { 2 0 0 8 }$ , LargeVis $\\left( \\mathbb { T a n g e t a l . } \\right) , \\left[ 2 0 1 6 \\right)$ and UMAP (McInnes et al., 2018) has been on preserving the local neighborhood structure of each individual point. Similarly, the common performance measures of DR such as trustworthinesscontinuity (Venna & Kaski, 2005), precision-recall (i.e. AUC) $\\left( \\mathrm { N e n n a ~ e t ~ a l . } \\right) \\left[ \\mathrm { 2 0 1 0 } \\right]$ , and nearestneighbor accuracy have also been developed by retaining the same focus on reflecting the local accuracy of the embedding. Thus, there has been a lack of attention on developing methods that focus on preserving the global structure of the data and likewise, practical performance measures to assess the global accuracy. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We first introduce the global score, a quantitative measure which reflects the closeness of a given embedding to the PCA embedding (which is optimal by means of preserving the data variance). The purpose of this score is to measure the accuracy of an embedding in reflecting the overall placement of the clusters of points relative to their original representation in high-dimension. By design, PCA yields the highest global score among all the DR methods and high values of global score indicates the efficacy of a DR method in reflecting the global structure. ",
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+        "type": "text",
+        "text": "Next, we introduce TriMap, a DR method which focuses on preserving the global structure of the data in the embedding. Pairwise (dis)similarities between points (used by the previous DR methods) seem to be insufficient in capturing the global structure. Instead, TriMap incorporates a higher order of structure to construct the embedding by means of triplets: ",
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+        "img_path": "images/469e54a291abb66f79d2ada5a5b788c0cdeb04f15d8756beeee1e35e115859b4.jpg",
+        "text": "$$\n( i , j , k ) \\Leftrightarrow p o i n t i s c l o s e r t o p o i n t j t h a n p o i n t k .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The key idea behind TriMap stems from semi-supervised metric learning $\\left( \\overline { { \\mathrm { A m i d e t a l . } } } , \\overline { { 2 0 1 6 } } \\right)$ : Given an initial low-dimensional representation for the data points, the triplet information from the highdimensional representation of the points is used to enhance the quality of the embedding. Similarly, TriMap is initialized with the low dimensional PCA embedding, and this embedding is then modified using a set of carefully selected triplets from the high-dimensional representation. ",
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+        "img_path": "images/126550d3d027ba058dc4c638816e25f31438dc6863e0896965d53054c49a4bed.jpg",
+        "image_caption": [
+            "Figure 1: 2-D Visualizations of the S-curve dataset: (a) original dataset in 3-D, (b) t-SNE, (c) UMAP, (d) TriMap, and (e) PCA. The values of AUC and global score, for respectively measuring local and global accuracy, are shown in order as a pair (AUC, GS) for each embedding. Despite having higher AUC values, t-SNE and UMAP both fail to reflect the overall shape of the S-curve. On the other hand, TriMap successfully unveils the underlying structure in the original dataset. Note that GS is the only DR performance measure that can reflect this property. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "With an extensive set of experiments, we show that TriMap produces excellent results on a variety of real-world as well as synthetic datasets. We show that in many cases TriMap outperforms all the competitor non-linear methods by means of global score and provides comparable local accuracy. While being significantly faster than t-SNE, TriMap provides comparable runtime to UMAP and LargeVis while scaling drastically better to larger datasets. On the Character Font Images dataset of ${ \\sim } 1 . 7 \\mathbf { M }$ points, TriMap calculates the embedding in ${ \\sim } 1 . 3$ hours while LargeVis takes more than 3 hours and UMAP exceeds the 12 hours time limit. Our contributions can be summarized as follows: ",
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+        "text": "• We introduce a global score to quantify the quality of a low-dimensional embedding in reflecting the global structure of the high-dimensional data such as placement of the clusters rather than the local neighborhood of individual points.   \nWe introduce TriMap, a fast dimensionality reduction method which provides embeddings of the data that are globally more accurate than other non-linear DR methods such as t-SNE, LargeVis, and UMAP.   \n• We provide an efficient implementation1 of TriMap that can easily scale to millions of points on commodity hardware and outperforms the competing methods in terms of runtime. We also perform many large-scale experiments on various datasets to show the efficacy of TriMap in terms of DR performance measures and runtime. ",
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+    {
+        "type": "text",
+        "text": "2 A MEASURE OF GLOBAL ACCURACY ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Consider the S-curve dataset2 which consists of 5000 points in 3-D uniformly sampled from an S-shaped manifold (Figure 1.(a)). This dataset serves as a paradigmatic problem for evaluating the performance of DR methods. In Figure $\\bigstar \\bigstar$ we show the results of 2-D embeddings of the S-curve dataset using t-SNE, UMAP, TriMap, and PCA. The top of each graph is labeled by the scoring pair (AUC, GS) where GS stands for global score (introduced below). Note that both t-SNE and UMAP provide higher values of the AUC score and locally preserve the continuity of the manifold. However, they both fail to recover the global structure of the S-curve, which is naturally reflected in the PCA embedding. On the other hand, our TriMap method (formally defined later) successfully recovers the structure of the S-curve by “unveiling” the curved shape of the manifold at both ends. Overall, the 2-D TriMap embedding resembles the original 3-D representation as much as possible. Note also that GS is the only measure that can reflect the global accuracy of the embedding. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The previous example indicates that the local measures of DR performance (such as AUC) cannot reflect the global accuracy of a low-dimensional embedding. In fact, the low-granular structure of the data can only be estimated by considering the global statistics of the dataset, as regarded by the PCA method. PCA is a linear DR method that projects the high-dimensional data onto the top- $d$ orthogonal directions having the highest variance. In order to calculate the mapping, PCA only considers the aggregate statistics of the dataset rather than the local information of each individual data point. As a result, PCA is extremely well suited at retaining the global structure of the data, i.e. the overall shape of the dataset, placement of the clusters, and existence of potential outliers. ",
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+        "img_path": "images/90058bdad23fe495c6d48369d44b01219a795fa12a3178643624027d56f79383.jpg",
+        "image_caption": [
+            "Figure 2: The Effect of the weight transformation on the MNIST dataset: (a) no weight transformation, (b) $\\gamma = 5 0$ , (c) $\\gamma = 5 0 0$ (default), and (d) $\\gamma = 5 0 0 0$ . The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure. Larger values of $\\gamma$ emphasizes more on the local accuracy rather than the global accuracy. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "However, by focusing on the global structure, PCA loses much of the local information such as the neighborhood structure of each data point. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Given a low-dimensional mapping produced by PCA, it is possible to calculate an optimal inverse mapping into the original high-dimensional space by means of minimizing the squared error. The optimal inverse map also corresponds to a linear mapping3. In order to quantify the global accuracy of a DR result, we focus on the accuracy of the embedding in reflecting the global structure of the data similar to PCA. That is, we consider the minimum reconstruction error of the original dataset by means of a linear inverse map. Given $n$ data points $\\{ \\pmb { x } _ { i } \\in \\mathbb { R } ^ { m } \\} _ { i = 1 } ^ { n }$ , let $\\pmb { X } \\in \\mathbb { R } ^ { m \\times \\bar { n } }$ denote the high-dimensional data matrix where the $i$ -th column corresponds to $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ . Similarly, let $\\pmb { Y } \\in \\mathbb { R } ^ { d \\times n }$ denote the matrix of the low-dimensional embedding of the points $\\{ \\pmb { y } _ { i } \\in \\mathbb { R } ^ { d } \\} _ { i = 1 } ^ { n }$ . Without loss of generality, we assume both $\\boldsymbol { X }$ and $\\mathbf { Y }$ are centered. We define the Minimum Reconstruction Error $( M R E )$ from the embedding as ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/70767c3053b6592c7e40d03f39c83a4609bd563f87500a9c23ae2c69dbbaaeb6.jpg",
+        "text": "$$\n\\mathcal { E } ( \\boldsymbol { Y } | \\boldsymbol { X } ) : = \\operatorname* { m i n } _ { \\boldsymbol { A } \\in \\mathbb { R } ^ { m \\times d } } \\| \\boldsymbol { X } - \\boldsymbol { A } \\boldsymbol { Y } \\| _ { \\mathrm { F } } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "where $\\| \\cdot \\| _ { \\mathrm { F } }$ denotes the Frobenius norm4. Note that PCA has the lowest possible MRE among all the DR methods. Thus, in order to obtain a normalized measure of global accuracy of a given embedding $\\mathbf { Y }$ for a data $\\boldsymbol { X }$ , we define the global score $( G S )$ as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/11ca6ad4d51185c5fc175b9cd72f52e3c8f252936418b98b494531efda9d9ece.jpg",
+        "text": "$$\n\\mathrm { G S } ( Y | X ) : = \\exp \\Big ( - \\frac { \\mathcal { E } ( Y | X ) - \\mathcal { E } _ { \\mathrm { P C A } } } { \\mathcal { E } _ { \\mathrm { P C A } } } \\Big ) \\in [ 0 , 1 ] ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { E } _ { \\mathrm { P C A } } : = \\mathcal { E } ( Y _ { \\mathrm { P C A } } | X )$ denotes the MRE achieved by the PCA embedding $\\mathbf { Y } _ { \\mathrm { P C A } }$ on the same dataset $\\boldsymbol { X }$ . Note that $\\mathrm { \\bf G S } ( Y _ { \\mathrm { P C A } } | X ) = 1$ and we claim that larger values of GS indicate a higher capacity of a DR method to reflect the global structure of the data, as shown in the experiments. ",
+        "bbox": [
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+        "type": "text",
+        "text": "In the remainder of the paper, we use GS as the global measure of performance. Due to the high computational complexity for calculating the trustworthiness-continuity and AUC scores for large data sets, we use nearest-neighbors accuracy as the local measure of performance henceforth. ",
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+    {
+        "type": "text",
+        "text": "3 THE TRIMAP METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We now formally introduce the TriMap method. Recall that a triplet consists of three points $( i , j , k )$ where point $i$ is closer to point $j$ than point $k$ . TriMap chooses a subset $\\mathcal { T } = \\{ ( i , j , \\bar { k } ) \\}$ of triplets and assigns a weight $\\omega _ { i j k } \\geq 0$ for each triplet: a higher value of $\\omega _ { i j k }$ implies that the pair $( i , k )$ is located much farther than the pair $( i , j )$ . We define the loss of the triplet $( i , j , k )$ as ",
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+        "img_path": "images/3e637a0bf691c66c546753c68ab6a681797eecb1431e08128a32ecf417f3c676.jpg",
+        "text": "$$\n\\ell _ { i j k } : = \\omega _ { i j k } \\frac { s ( y _ { i } , y _ { k } ) } { s ( y _ { i } , y _ { j } ) + s ( y _ { i } , y _ { k } ) } , \\mathrm { ~ w h e r e ~ } s ( y _ { i } , y _ { j } ) = \\left( 1 + | y _ { i } - y _ { j } | | ^ { 2 } \\right) ^ { - 1 } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "image",
+        "img_path": "images/922bb546e53ad0604be26eae945edea542602a5fe252f6a85aabb42a6bfac050.jpg",
+        "image_caption": [
+            "Figure 3: Effect of changing the number of triplets on the quality of the embeddings of the MNIST dataset. We consider $( m , m ^ { \\prime } , r ) = c \\times ( 2 , 1 , 1 )$ for: (a) $c = 1$ , (b) $c = 2$ , (c) $c = 5$ (default), (d) $c = 1 0$ , and (e) $c = 2 0$ . The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure. The quality of embedding does not improve significantly after adding a certain number of triplets. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "is a similarity function between $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { y } } _ { i } }$ and ${ \\pmb y } _ { j }$ . The choice of $s$ is motivated by the good performance of Student t-distribution for similarities in low-dimension in the t-SNE method. Note that the loss of the triplet $( i , j , k )$ approaches zero as $\\| \\pmb { y } _ { i } - \\pmb { y } _ { j } \\|$ decreases and $\\| \\pmb { y } _ { i } - \\pmb { y } _ { k } \\|$ increases. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first develop the weighing scheme for the triplets. To reflect the relative similarities in high-dimension, we define the unnormalized weight of the triplet $( i , j , k )$ as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/60e1e83cb9917f7a3ad4874b7b811e293f1233316e92ff44fedf8417d515f808.jpg",
+        "text": "$$\n\\tilde { \\omega } _ { i j k } = \\exp ( d _ { i k } ^ { 2 } - d _ { i j } ^ { 2 } ) \\geq 0 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "in which, $d _ { i j }$ is any distance measure between $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ and $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { j } }$ in highdimension. For Euclidean distances, we use the scaling introduced in (Zelnik-Manor & Perona, $\\textcircled { 2 0 0 5 }$ , ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/b504e789e0a052a54ad75c5ee05faf1406738ff10769b5c0faff0f78f976f39a.jpg",
+        "text": "$$\nd _ { i j } ^ { 2 } = \\frac { \\lVert \\pmb { x } _ { i } - \\pmb { x } _ { j } \\rVert ^ { 2 } } { \\sigma _ { i j } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma _ { i j } = \\sigma _ { i } \\sigma _ { j }$ and $\\sigma _ { i }$ is set to the average Euclidean distance between $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ and the set of nearest-neighbors of $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ from 4-th to 6-th neighbors. This choice of $\\sigma _ { i j }$ adaptively adjusts the scaling based on the density of the data. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/ccbf9f3fa2e2973959a501e01799674558f18b1188314acb65e1070afdf301e8.jpg",
+        "image_caption": [
+            "Figure 4: $\\gamma$ -scaled logtransformation with different values of $\\gamma$ . The value NIL corresponds to no transformation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While the choice of weights $\\tilde { \\omega } _ { i j k }$ works well in practice, we adjust the weights further by applying a non-linear transformation that emphasizes the smaller weights. Expanding the values of small weights has the effect of placing the nearest-neighbors closer to the point and pushing the remaining points farther away, thus improving the local accuracy (as shown in Figure $\\bigtriangledown$ and discussed later). The final value of the weight $\\omega _ { i j k }$ is obtained by applying the $\\gamma$ -scaled log-transformation (see Figure 4), ",
+        "bbox": [
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "equation",
+        "img_path": "images/c682dc70f91fa4a210a18a9983e5a9365a49bbbf9847bdb5324b218467962940.jpg",
+        "text": "$$\n\\omega _ { i j k } = \\zeta _ { \\gamma } \\big ( \\frac { \\tilde { \\omega } _ { i j k } } { \\gamma } + \\delta \\big ) \\quad \\mathrm { w h e r e } \\zeta _ { \\gamma } ( u ) : = \\log \\big ( 1 + \\gamma u \\big ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "in which $\\mathcal { W } = \\mathrm { m a x } _ { ( i ^ { \\prime } , j ^ { \\prime } , k ^ { \\prime } ) \\in \\mathcal { T } } \\tilde { \\omega } _ { i ^ { \\prime } j ^ { \\prime } k ^ { \\prime } } , ~ \\gamma > 0$ is a scaling factor, and $\\delta$ is a small constant. We use $\\gamma = 5 0 0$ and $\\delta = 1 0 ^ { - 4 }$ in all our experiments. ",
+        "bbox": [
+            173,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To construct the embedding, we consider a small subset of all possible triplets $( i , j , k )$ for which, the closer point $j$ belongs to the set of nearest-neighbors of the point $i$ and the farther point $k$ is among the points that are more distant from $i$ than $j$ , chosen uniformly at random. For each point we consider its $m = 1 0$ nearest neighbors and sample $m ^ { \\prime } = 5$ triplets per nearest-neighbor. This yields $m \\times m ^ { \\prime } = 5 0$ nearest-neighbor triplets per point. In addition, we also add $r = 5$ random triplets $( i , j , k )$ per each point $i$ where $j$ and $k$ are sampled uniformly at random and their order is possibly switched based on their nearness to $i$ . This yields $m \\times m ^ { \\prime } + r = 5 5$ triplets per point in total. Thus, the overall complexity of the optimization step is linear in number of points $n$ . The computational complexity is dominated by the nearest-neighbor search, which is shared among all the recent methods such as t-SNE, LargeVis, and UMAP. We use ANNOY for the approximate nearest-neighbor search5 which is based on random projection trees. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "While a random initialization for the embedding also works well in practice, we initialize the embedding to the PCA solution $\\mathbf { Y } _ { \\mathrm { P C A } }$ (scaled by a small constant value for better convergence). The PCA initialization for TriMap allows faster convergence while preserving much of the global structure discovered by PCA. Note that the other DR methods such as t-SNE are extremely sensitive to the initialization and do not converge well with any initial solution other than small random initialization around the origin. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/db27f423ec334add935e2b9913ed85f659e9bcf69152030db39ac67779d1a70c.jpg",
+        "table_caption": [
+            "Table 1: Runtime of the methods in hh:mm:ss format on single machine with $2 . 6 \\ : \\mathrm { G H z }$ Intel Core i5 CPU and 16 GB of memory. We limit the runtime of each method to 12 hours. Also, UMAP runs out of memory on datasets larger than ${ \\sim } 4 \\mathbf { M }$ points. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset (size)</td><td>t-SNE</td><td>LargeVis</td><td>UMAP</td><td>TriMap</td><td>Speedup</td></tr><tr><td>COIL-20 (1440)</td><td>00:00:08</td><td>00:05:51</td><td>00:00:04</td><td>00:00:02</td><td>2.00×</td></tr><tr><td>USPS (11K)</td><td>00:02:02</td><td>00:06:12</td><td>00:00:12</td><td>00:00:11</td><td>1.10×</td></tr><tr><td>Epileptic Seizure (11.5K)</td><td>00:03:11</td><td>00:06:17</td><td>00:00:15</td><td>00:00:12</td><td>1.25×</td></tr><tr><td>20 Newsgroup (18K)</td><td>00:05:34</td><td>00:06:57</td><td>00:00:26</td><td>00:00:21</td><td>1.24×</td></tr><tr><td>TabulaMuris (54K)</td><td>00:17:32</td><td>00:09:29</td><td>00:01:12</td><td>00:01:06</td><td>2.00×</td></tr><tr><td>MNIST(70K)</td><td>00:20:38</td><td>00:11:29</td><td>00:01:15</td><td>00:01:23</td><td>0.90×</td></tr><tr><td>FashionMNIST(70K)</td><td>00:19:10</td><td>00:11:04</td><td>00:01:18</td><td>00:01:24</td><td>0.93×</td></tr><tr><td>TVNews (~129K)</td><td>00:38:59</td><td>00:16:26</td><td>00:02:57</td><td>00:02:45</td><td>1.07×</td></tr><tr><td>360+KLyrics (~360K)</td><td>08:50:49</td><td>00:44:16</td><td>00:25:23</td><td>00:13:49</td><td>1.84×</td></tr><tr><td>Covertype(~581K)</td><td>二</td><td>00:44:54</td><td>02:59:41</td><td>00:24:42</td><td>1.82×</td></tr><tr><td>RCV1 (800K)</td><td>二</td><td>01:34:38</td><td>04:55:53</td><td>00:36:59</td><td>2.56×</td></tr><tr><td>Character Font Images (~1.7M)</td><td>二</td><td>03:16:19</td><td>二</td><td>01:17:50</td><td>2.52×</td></tr><tr><td>KDDCup99 (~4.9M)</td><td>二</td><td>二</td><td>二</td><td>04:17:01</td><td>二</td></tr><tr><td>HIGGS (11M)</td><td>1</td><td>1</td><td>1</td><td>10:08:36</td><td>1</td></tr></table>",
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+        "text": "We define the final loss as the sum of the losses of the sampled triplets in $\\tau$ ",
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+    {
+        "type": "equation",
+        "img_path": "images/d5980b124afe5e0c24de92f85002804d0e39c51da386df61b04fbee968c41935.jpg",
+        "text": "$$\n\\ell _ { \\mathrm { T r i M a p } } = \\sum _ { ( i , j , k ) \\in \\mathcal { T } } \\ell _ { i j k } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The loss is minimized using the full-batch gradient descent with momentum using the delta-bar-delta method. In all our experiments, we perform 400 iterations with the value of momentum parameter equal to 0.5 during the first 250 iterations and 0.8 afterwards. ",
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+        "type": "text",
+        "text": "Finally, note that there exists connections between TriMap and a number of triplet (aka ordinal) embedding methods such as t-STE (Van Der Maaten & Weinberger, $\\boxed { 2 0 1 2 } )$ . The triplet embedding methods have been developed for a different setting where the goal is to find an embedding based on a given pre-specified set of triplets obtained from human evaluators (or some form of implicit feedback). For instance, t-STE maximizes the sum of log of the satisfaction probabilities of the triplets to calculate the embedding. It is worth mentioning that TriMap is a DR method that is designed to sample the informative triplets from the high-dimensional representation of a set of points and assign weights to these triplets to reflect the relative similarities of these points. Although TriMap can also be used for the triplet embedding task, we only focus on the DR results 6. ",
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+    {
+        "type": "text",
+        "text": "3.1 EFFECT OF DIFFERENT PARAMETERS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We briefly discuss the effect of different parameters, namely the total number of triplets $| \\tau |$ and the $\\gamma$ -scaled log-transformation, on the quality of the embedding. TriMap is particularly robust to the number of sampled triplet for constructing the embedding. This can be explained by the high amount of redundancy in the triplets (the triplets $( i , j , k )$ and $\\bar { ( } i , j , k ^ { \\prime } )$ convey the same information if $k$ and $k ^ { \\prime }$ are nearest neighbors and also mapped nearby). In Figure $\\textcircled { 3 }$ we consider various values for $m$ , $m ^ { \\prime }$ , and $r$ for the MNIST dataset while fixing the remaining parameters. In fact, using large number of triplets can sometimes introduce an overhead and require larger number of iterations to converge. ",
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+        "text": "A more important parameter is $\\gamma$ which controls the trade-off between the local and global accuracy. Larger values of $\\gamma$ increases the relative importance of triplets with smaller weights. This causes the method to focus on the nearest-neighbor points rather than the points that are far away, thus improving the local accuracy. On the other hand, improving the local accuracy can impair the global accuracy. In Figure $^ { 2 , }$ we plot the $\\gamma$ -scaled log-transformation for various $\\gamma$ values and illustrate the results on MNIST without the log-transformation as well as the results with different $\\gamma$ values. For larger values of $\\gamma$ the clusters tend to become more compressed and as a result, the nearest-neighbor accuracy is improved. On the other hand, the global score starts to decrease for larger $\\gamma$ values. ",
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+        "img_path": "images/2ce1b1fc67b1b7febaf375f4781a2a98d8cfd034c15252995fbc93981b425354.jpg",
+        "image_caption": [
+            "Figure 5: Visualizations of different datasets using t-SNE, UMAP, TriMap, and PCA. Each row corresponds to one dataset and each column represents one method. The values of nearest neighbor accuracy and global score are shown as a pair (NN,GS) on top of each figure. "
+        ],
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+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "In this section, we apply TriMap on a set of real-world as well as synthetic datasets and compare the results to t-SNE, LargeVis, UMAP, and PCA methods. The datasets used in our experiments are listed in Table 1 and a short description is given in the appendix. All experiments are conducted on a single machine with $2 . 6 \\ : \\mathrm { G H z }$ Intel Core i5 CPU and $1 6 \\mathrm { \\ G B }$ of memory. We limit the runtime of each algorithm to 12 hours. For implementations, we use the default sklearn implementation for t-SNE and the official implementations of LargeVis and UMAP provided by the authors7,8. Due to lack of space, we provide the comparison to the LargeVis results as well additional TriMap results on the larger datasets in the appendix. ",
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+        "img_path": "images/c30bd6f21a0a26542831274847c6c7fa12491842072df61b0d6c9e9ef3c98985.jpg",
+        "image_caption": [
+            "Figure 5: Visualizations of different datasets (continued) using t-SNE, UMAP, TriMap, and PCA. Each row corresponds to one dataset and each column represents one method. The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure. "
+        ],
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+        "type": "text",
+        "text": "In order to have a fair comparison, we use the default parameter values for all methods, including ours $\\mathbf { \\Phi } _ { m } = 1 0 \\mathbf { \\Phi } $ , $m ^ { \\prime } = 5$ , $r \\ = \\ 5$ , $\\gamma ~ = ~ 5 0 0$ , and 400 iterations). Also to reduce the overhead induced by the dimensionality of the data in the nearest-neighbor search step, we reduce the number of dimensions of the dataset to 100 if necessary, using the PCA method. To evaluate the local performance, we show the nearest-neighbor accuracy of each result. We also show the GS as a measure of global performance. The performance measures are shown on top of each figure as a pair (NN, GS). ",
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+    {
+        "type": "text",
+        "text": "4.1 RUNTIME ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The runtime of the methods are provided in Table $^ 1$ in the hh:mm:ss format. We limit the runtime of each method to 12 hours. As can be seen from the results, TriMap provides excellent runtime and outperforms all the other methods in most cases. Also, TriMap easily scales to millions of points while the other methods exceed the time limit or run out of memory. For instance, UMAP causes an out of memory error for datasets larger than ${ \\sim } 4 \\mathbf { M }$ points. ",
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+    {
+        "type": "text",
+        "text": "4.2 VISUALIZATIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The visualizations of the datasets using TriMap as well as the other competing methods are shown in Figure $5$ and $\\bigtriangledown$ For some results, we provide a zoomed in snippet over the main figure to provide a more detailed illustration. Overall, TriMap preserves the underlying global structure of the data better than the other competing methods. This is reflected by the larger GS values for TriMap as well as visually comparing the embeddings to the PCA result. For example, TriMap recovers the continuous structure of the TV news dataset and separates the remaining outliers in the data which are also identified by the PCA method. This can be verified by comparing the placement of an example outlier point, marked with a red $\\times$ , by the different methods: TriMap shows this point among other outliers whereas t-SNE and UMAP fail to uncover this information. Also, the global score of TriMap on this dataset is much higher than the other methods. Further discussion is given in the appendix. ",
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+        "image_caption": [
+            "Figure 6: Visualizations of Covertype and RCV1 datasets using UMAP, TriMap, and PCA, and visualizations of the Character Font Images dataset using LargeVis, TriMap, and PCA. The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure. "
+        ],
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "TriMap is a fast and efficient method that can be easily applied to large datasets. While TriMap is extremely effective for uncovering the global structure of the data, other methods such as t-SNE can provide additional insight about the local neighborhood of individual points. As a future research direction, we consider using pairwise constraints along with triplet constraints to improve the local accuracy. The current implementation of TriMap utilizes a single core. Parallel implementation of the method that can exploit multiple cores is another future direction. Furthermore, the global accuracy is measured in terms of the global score which is based on the assumption that linear projection obtained by PCA is globally optimal. While our global score can provide insight about the global accuracy of the embedding in many cases, it appears to be ineffective when the data is highly non-linear or contains a large amount of outliers. Developing non-linear and more robust global performance measures could significantly improve the assessment of the DR results and provide guidelines for developing more accurate DR techniques. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ehsan Amid, Aristides Gionis, and Antti Ukkonen. Semi-supervised kernel metric learning using relative comparisons. arXiv preprint arXiv:1612.00086, 2016.   \nLaurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(Nov):2579–2605, 2008.   \nLeland McInnes, John Healy, and James Melville. Umap: Uniform manifold approximation and projection for dimension reduction. arXiv preprint arXiv:1802.03426, 2018.   \nKarl Pearson. Liii. on lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11):559–572, 1901.   \nVin D Silva and Joshua B Tenenbaum. Global versus local methods in nonlinear dimensionality reduction. In Advances in neural information processing systems, pp. 721–728, 2003.   \nJian Tang, Jingzhou Liu, Ming Zhang, and Qiaozhu Mei. Visualizing large-scale and highdimensional data. In Proceedings of the 25th international conference on world wide web, pp. 287–297. International World Wide Web Conferences Steering Committee, 2016.   \nLaurens Van Der Maaten and Kilian Weinberger. Stochastic triplet embedding. In 2012 IEEE International Workshop on Machine Learning for Signal Processing, pp. 1–6. IEEE, 2012.   \nJarkko Venna and Samuel Kaski. Local multidimensional scaling with controlled tradeoff between trustworthiness and continuity. In Proceedings of 5th Workshop on Self-Organizing Maps, pp. 695–702. Citeseer, 2005.   \nJarkko Venna, Jaakko Peltonen, Kristian Nybo, Helena Aidos, and Samuel Kaski. Information retrieval perspective to nonlinear dimensionality reduction for data visualization. Journal of Machine Learning Research, 11(Feb):451–490, 2010.   \nLihi Zelnik-Manor and Pietro Perona. Self-tuning spectral clustering. In Advances in neural information processing systems, pp. 1601–1608, 2005. ",
+        "bbox": [
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+            125,
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+            541
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "A DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+            102,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The datasets used in the experiments are listed below. All datasets are publicly available online and a download link is provided. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "• COIL- ${ \\bf \\nabla } \\cdot 2 { \\bf 0 } ^ { 9 }$ (1440): gray-scale images of 20 objects in uniformly sampled orientations (5 degrees of rotation, 72 images per object). Each image is pre-processed by having the background removed and cropped into size $1 2 8 \\times 1 2 8$ . $\\mathbf { U S P S } ^ { 1 0 }$ (11K): images of handwritten digits (0–9) of size $1 6 \\times 1 6$ . Epileptic Seizure11 (11.5K): EEG signal recordings of brain activity for seizure recognition. It contains 178-dimensional vectors belonging to 5 categories.   \n20 Newsgroup11 (18K): newsgroup posts categorized into 20 topics. We use a TF-IDF representation of the words in each document as the features.   \nTabula Muris12 $( \\sim 5 4 \\mathrm { K } )$ : single cell transcriptome data from the mouse from 20 organs.   \n• MNIST13 (70K): images of handwritten digits (0–9) of size $2 8 \\times 2 8$ .   \n• Fashion MNIST14 (70K): gray-scale images of clothing items such as t-shirt, pullover, bag, etc. of size $2 8 \\times 2 8$ .   \nTV News11 $( \\sim 1 2 9 \\mathrm { K } )$ : audio-visual features from TV news broadcast categorized into commercial and non-commercial.   \n• $\\mathbf { 3 6 0 K + }$ Lyrics15 $( \\sim 3 6 2 \\mathrm { K } )$ : lyrics of songs from 12 different genres. We group similar genres together (metal-rock, R&B-pop, etc.) to form 7 groups. We use the TF-IDF representation of the words in the song as the features. Covertype11 $( \\sim 5 8 1 \\mathrm { K } )$ : cartographic features for forest cover type prediction.   \n• $\\mathbf { R C V 1 } ^ { 1 6 }$ (800K): Reuters Corpus Volume I archive of categorized newswire stories. Character Font Images11 $( { \\sim } 1 . 7 \\mathrm { M } )$ : images of character from scanned and computer generated fonts.   \n• $\\mathbf { K D D C u p 9 9 } ^ { \\mathrm { 1 1 } }$ $( { \\sim } 4 . 9 \\mathrm { { M } ) }$ : computer network intrusion detection.   \n• $\\mathbf { H } \\mathbf { I } \\mathbf { G } \\mathbf { G } \\mathbf { S } ^ { 1 1 }$ (11M): Higgs bosons recognition from a background process. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B MORE VISUALIZATIONS ",
+        "text_level": 1,
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "We compare the results of TriMap to LargeVis in Figure 7 and 8. We also provide more visualizations obtained using TriMap in Figure 9. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "C DISCUSSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We briefly discuss the results of TriMap and draw a comparison to the other methods. ",
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+        "type": "text",
+        "text": "TriMap generally provides better global accuracy compared to the competing methods. It also successfully maintains the continuity of the underlying manifold. This can be seen from the COIL-20 result where certain clusters are located farther away from the remaining clusters. However, the underlying structure for the main cluster resembles the one provided by the other methods. TriMap also preserves the continuous structure in the Fashion MNIST and the TV News datasets. ",
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+    {
+        "type": "text",
+        "text": "TriMap is also efficient in uncovering the possible outliers in the data. For instance, PCA reveals a large number of outliers in the Tabula Muris and the $3 6 0 \\mathrm { + K }$ Lyrics datasets. These outliers are located far away from the main clusters in the TriMap results. However, the same points are located very close to the remaining points in the t-SNE results. ",
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+    {
+        "type": "table",
+        "img_path": "images/dfe3f3a202b26c6ae5fdc9d551543847cb35a69d02309223d6b5682470f72062.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>http://www.cs.columbia.edu/CAvE/software/softlib/coil-20.php</td></tr><tr><td>https://www.kaggle.com/bistaumanga/usps-dataset</td></tr><tr><td>http://archive.ics.uci.edu/ml/index.php</td></tr><tr><td>https://tabula-muris.ds.czbiohub.org/</td></tr><tr><td>http://yann.lecun.com/exdb/mnist/</td></tr><tr><td>https://github.com/zalandoresearch/fashion-mnist</td></tr><tr><td>https://www.kaggle.com/gyani95/380000-lyrics-from-metrolyrics</td></tr><tr><td>https://scikit-learn.org/0.18/datasets/rcv1.html</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/1b3d08ee2fd1e362cbb60fdc07451f5e9efbec7336f6557e5cc45d24a6ff6924.jpg",
+        "image_caption": [
+            "Figure 7: Visualizations of different datasets using LargeVis, TriMap, and PCA. Each row corresponds to one dataset and each column represents one method. The values of nearest neighbor accuracy and global score are shown as a pair (NN,GS) on top of each figure. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/02894110afa287f8baca3934fc4118784cc7dec51a228d2a276abdd1c244e9a6.jpg",
+        "image_caption": [
+            "Figure 7: Visualizations of different datasets (continued) using LargeVis, TriMap, and PCA. Each row corresponds to one dataset and each column represents one method. The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Additionally, both t-SNE and LargeVis tend to form spurious clusters by splitting the underlying connected manifold. This can be seen from the TV News results and the result of LargeVis on the Covertype dataset. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Finally, notice that in some cases GS fails to reflect the global accuracy of the embeddings. This can be seen from the low GS values for all methods on the Covertype dataset. GS may become uninformative when there exists a high degree of non-linearity in the data that cannot be reflected using PCA. GS also cannot reflect the accuracy of the embedding in uncovering single outliers. Developing more accurate global measures for these scenarios is a future research direction. ",
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+        "img_path": "images/cf9669f5038690e85b2e3c5971d63271a86fb8c99aa4da0cb5ee4f8f808801dd.jpg",
+        "image_caption": [
+            "Figure 8: Visualizations of Covertype and RCV1 datasets using LargeVis, TriMap, and PCA, and visualizations of the Character Font Images dataset using LargeVis, TriMap, and PCA. The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
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+        "img_path": "images/3861d063bab8a11cd7e97eb87ff420798a417be26fec70be2875a5b800d7374e.jpg",
+        "image_caption": [
+            "Figure 9: Visualizations of KKDCup99 and HIGGS datasets TriMap and PCA. Each row corresponds to one dataset and each column represents one method. The values of nearest neighbor accuracy and global score are shown as a tuple (NN,GS) on top of each figure. TriMap shows more structure for both datasets than PCA. Note that GS is uninformative for the KDDCup99 dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 12
+    }
+]

parse/train/H1efEp4Yvr/H1efEp4Yvr.md

@@ -0,0 +1,517 @@
+# GLOBAL CONCAVITY AND OPTIMIZATION IN A CLASS OF DYNAMIC DISCRETE CHOICE MODELS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Discrete choice models with unobserved heterogeneity are commonly used Econometric models for dynamic Economic behavior which have been adopted in practice to predict behavior of individuals and firms from schooling and job choices to strategic decisions in market competition. These models feature optimizing agents who choose among a finite set of options in a sequence of periods and receive choice-specific payoffs that depend on both variables that are observed by the agent and recorded in the data and variables that are only observed by the agent but not recorded in the data. Existing work in Econometrics assumes that optimizing agents are fully rational and requires finding a functional fixed point to find the optimal policy. We show that in an important class of discrete choice models the value function is globally concave in the policy. That means that simple algorithms that do not require fixed point computation, such as the policy gradient algorithm, globally converge to the optimal policy. This finding can both be used to relax behavioral assumption regarding the optimizing agents and to facilitate Econometric analysis of dynamic behavior. In particular, we demonstrate significant computational advantages in using a simple implementation policy gradient algorithm over existing “nested fixed point” algorithms used in Econometrics.
+
+# 1 INTRODUCTION
+
+Dynamic discrete choice model with unobserved heterogeneity is, arguably, the most popular model that is currently used for Econometric analysis of dynamic behavior of individuals and firms in Economics and Marketing (e.g. see surveys in Eckstein and Wolpin (1989), Dubé et al. (2002) Abbring and Heckman (2007), Aguirregabiria and Mira (2010)). Even most recent Econometric papers on single-agent dynamic decision-making use this setup to showcase their results (e.g. Arcidiacono and Miller, 2011; Aguirregabiria and Magesan, 2016; Müller and Reich, 2018).In this model, pioneered in Rust (1987), the agent chooses between a discrete set of options (typically 2) in a sequence of discrete time periods to maximize the expected cumulative discounted payoff. The reward in each period is a function of the state variable which follows a Markov process and is observed in the data and also a function of an idiosyncratic random variable that is only observed by the agent but is not reported in the data. The unobserved idiosyncratic component is designed to reflect heterogeneity of agents that may value the same choice differently.
+
+Despite significant empirical success in prediction of dynamic economic behavior under uncertainty, dynamic discrete choice models frequently lead to seemingly unrealistic optimization problems that economic agents need to solve. For instance, Hendel and Nevo (2006) features an elaborate functional fixed point problem with constraints, which is computationally intensive, especially in continuous state spaces, for consumers to buy laundry detergent in the supermarket. Common approach for this functional fixed point problem is value function iteration (See Section 2.3 for more discussion).
+
+At the same time, rich literature on Markov Decision Processes (cf. Sutton and Barto, 2018) have developed several effective optimization algorithms, such as the policy gradient algorithm and its variants, that do not require solving for a functional fixed point. However, the drawback of the policy gradient is that the value function in a generic Markov Decision problem is not concave in the policy. This means that gradient-based algorithms have no guarantees for global convergence for a generic MDP. While for some specific and simple models where closed-form characterizations exist, the convergence results are shown by model-specific technique which is hard to generalize (e.g. Fazel et al., 2018, for linear quadratic regulator).
+
+In this paper our main goal is to resolve the dichotomy in empirical social science literature that the rationality of consumers requires for them to be able to solve the functional fixed point problem which is computationally intensive. Our main theoretic contribution is the proof that, in the class of dynamic discrete choice models with unobserved heterogeneity, the value function of the optimizing agent is globally concave in the policy. This implies that a large set of policy gradient algorithms that have a modest computational power requirement for the optimizing agents have a fast convergence guarantee in our considered class of dynamic discrete choice models. The importance of this result is twofold.
+
+First, it gives a promise that seemingly complicated dynamic optimization problems faced by consumers can be solved by relatively simple algorithms that do not require fixed point computation or functional optimization. This means that the policy gradient-style methods have an important behavioral interpretation. As a result, consumer behavior following policy gradient can serve as a behavioral assumption for estimating consumer preferences from data which is more natural for consumer choice settings than other assumptions that have been used in the past for estimation of preferences (e.g. -regret learning in Nekipelov et al. (2015)). Second, more importantly, our result showing fast convergence of the policy gradient algorithm makes it an attractive alternative to the search for the functional fixed point in this class of problems. While the goal of the Econometric analysis of the data from dynamically optimizing consumers is to estimate consumer preferences by maximizing the likelihood function, it requires to sequentially solve the dynamic optimization problem for each value of utility parameters along the parameter search path. Existing work in Economics prescribes to use fixed point iterations for the value function to solve the dynamic optimization problem (see Rust (1987), Aguirregabiria and Mira (2007)). The replacement of the fixed point iterations with the policy gradient method significantly speeds up the maximization of the likelihood function. This makes the policy gradient algorithm our recommended approach for use in Econometric analysis, and establishes practical relevance of many newer reinforcement learning algorithms from behavioral perspective for social sciences.
+
+# 2 PRELIMINARIES
+
+In this section, we introduce the concepts of the Markov decision process (MDP) with choice-specific payoff heterogeneity, the conditional choice probability (CCP) representation and the policy gradient algorithm.
+
+# 2.1 MARKOV DECISION PROCESS
+
+A discrete-time Markov decision process (MDP) with choice-specific heterogeneity is defined as a 5-tuple $\langle S , { \mathcal { A } } , r , \epsilon , { \mathcal { P } } , \beta \rangle$ , where $s$ is compact convex state space with $\mathrm { d i a m } ( \breve { S } ) \leq \breve { \bar { S } } < \infty$ , $\mathcal { A }$ is the set of actions, $r : \mathcal { S } \times \mathcal { A } \to \mathbb { R } _ { + }$ is the reward function, such that $r ( s , a )$ is the immediate non-negative reward for the state-action pair $( s , a )$ , $\epsilon$ are independent random variables, $\mathcal { P }$ is a Markov transition model where where $p ( s ^ { \prime } | s , a )$ defines the transition density between state $s$ and $s ^ { \prime }$ under action $a$ , and $\beta \in [ 0 , 1 )$ is the discount factor for future payoff. We assume that random variables $\epsilon$ are observed by the optimizing agent and not recorded in the data. These variables reflect idiosyncratic differences in preferences of different optimizing agents over choices. In the following discussion we refer to these variables as “random choice-specific shocks."
+
+In each period ${ t = 1 , 2 , \ldots , \infty }$ , the nature realizes the current state $s _ { t }$ based on the Markov transition $\mathcal { P }$ given the state-action pair $( s _ { t - 1 } , a _ { t - 1 } )$ in the previous period $t - 1$ , and the choice-specific shocks $\epsilon _ { t } \overset { \mathbf { \bar { \rho } } } { = } \{ \epsilon _ { t , a } \} _ { a \in \mathcal { A } }$ drawn i.i.d. from distribution $\epsilon$ . The optimizing agent chooses an action $a \in { \mathcal { A } }$ , and her current period payoff is sum of the immediate reward and the choice-specific shock, i.e., $\boldsymbol { \epsilon } _ { t , a }$ . Given initial state $s _ { 1 }$ , the agent’s long-term payoff is $\begin{array} { r } { \mathbf { E } _ { \epsilon _ { 1 } , s _ { 2 } , \epsilon _ { 2 } , \dots } \bigl [ \sum _ { t = 1 } ^ { \infty } \beta ^ { t - 1 } r ( s _ { t } , a _ { t } ) + \epsilon _ { t , a _ { t } } \bigr ] } \end{array}$ $r ( s , a ) +$ This expression makes it clear that random shocks $\epsilon$ play a crucial role in this model by allowing us to define the ex ante value function of the optimizing agent which reflects the expected reward from agent’s choices before the agent observes realization of $\epsilon _ { t }$ . When the distribution of shocks $\epsilon$ is sufficiently smooth (differentiable), the corresponding ex ante value function is smooth (differentiable)
+
+as well. This allows us to characterize the impact of agent’s policy on the expected value by considering functional derivatives of the value function with respect to the policy.
+
+In the remainder of the paper, we rely on the following assumptions.
+
+Assumption 2.1. The state space $s$ is compact in $\mathbb { R }$ and the action space $\mathcal { A }$ is binary, i.e., $\mathcal { A } = \{ 0 , 1 \}$
+
+Assumption 2.2. For all states s, the immediate reward $r ( s , 0 )$ for the state-action pair $( s , 0 )$ is zero i.e., $r ( s , 0 ) = 0$ , and the immediate reward $r ( s , 1 )$ for the state-action pair $( s , 1 )$ is bounded between $[ R _ { \mathrm { m i n } } , R _ { \mathrm { m a x } } ]$ .
+
+Assumption 2.3. Choice-specific shocks  are Type I Extreme Value random variables with location parameter $O$ (cf. Hotz and Miller, 1993) which are independent over choices and time periods.
+
+Assumption 2.1, 2.2, 2.3 are present in most of the papers on dynamic decision-making in economics, marketing and finance, (e.g. Dubé et al., 2002; Aguirregabiria and Mira, 2010; Arcidiacono and Miller, 2011; Aguirregabiria and Magesan, 2016; Müller and Reich, 2018)
+
+The policy and the value function A stationary Markov policy is a function $\sigma : \mathcal { S } \times \mathbb { R } ^ { A } \to \mathcal { A }$ which maps the current state $s$ and choice-specific shock $\epsilon$ to an action. In our further discussion we will show that there is a natural more restricted definition of the set of all feasible policies in this model.
+
+Given any stationary Markov policy $\sigma$ , the value function $V _ { \sigma } : { \mathcal { S } }  \mathbb { R }$ is a mapping from the initial state to the long-term payoff under policy $\sigma$ , i.e.,
+
+$$
+V _ { \sigma } ( s _ { 1 } ) = \mathbf { E } _ { \epsilon _ { 1 } , s _ { 2 } , \epsilon _ { 2 } , \ldots } \left[ \sum _ { t = 1 } ^ { \infty } \beta ^ { t - 1 } \left\{ r ( s _ { t } , \sigma ( s _ { t } , \epsilon _ { t } ) ) + \epsilon _ { t , \sigma ( s _ { t } , \epsilon _ { t } ) } \right\} \right] .
+$$
+
+Since the reward is non-negative and bounded, and the discount $\beta \in [ 0 , 1 )$ , value function $V _ { \sigma }$ is well-defined and the optimal policy $\tilde { \sigma }$ (i.e., $V _ { \tilde { \sigma } } ( s ) \geq V _ { \sigma } ( s )$ for all policies $\sigma$ and states $s$ ) exists. Furthermore, the following Bellman equation holds
+
+$$
+V _ { \sigma } ( s ) = \mathbf { E } _ { \epsilon } \bigl [ r ( s , \sigma ( s , \epsilon ) ) + \epsilon _ { \sigma ( s , \epsilon ) } + \beta \mathbf { E } _ { s ^ { \prime } } [ V _ { \sigma } ( s ^ { \prime } ) | s , \sigma ( s , \epsilon ) ] \bigr ] \qquad \mathrm { f o r ~ a l l ~ p o l i c i e s ~ } \sigma _ { \sigma ( s , \epsilon ) } .
+$$
+
+# 2.2 CONDITIONAL CHOICE PROBABILITY REPRESENTATION
+
+Based on the Bellman equation (1) evaluated at the optimal policy, the optimal Conditional Choice Probability $\tilde { \delta } ( a | s )$ (i.e., the probability of choosing action $a$ given state $s$ in the optimal policy $\tilde { \sigma }$ ) can be defined as
+
+$$
+\begin{array} { r } { \tilde { \delta } ( a | s ) = \mathbf { E } _ { \varepsilon } [ \mathbb { I } \{ r ( s , a ) + \epsilon _ { a } + \beta \mathbf { E } _ { s ^ { \prime } } [ V _ { \overline { { \sigma } } } ( s ^ { \prime } ) | s , a ] \geq r ( s , a ^ { \prime } ) + \epsilon _ { a ^ { \prime } } + \beta \mathbf { E } _ { s ^ { \prime } } [ V _ { \overline { { \sigma } } } ( s ^ { \prime } ) | s , a ^ { \prime } ] , \forall a ^ { \prime } \} ] } \end{array}
+$$
+
+The optimal policy $\tilde { \sigma }$ can, therefore, be equivalently characterized by threshold function ${ \tilde { \pi } } ( s , a ) =$ $r ( s , a ) + \beta \bar { \bf E _ { \tilde { s } ^ { \prime } } } [ \bar { V _ { \tilde { \sigma } } } ( s ^ { \prime } ) \vert s , a ]$ , such that the optimizing agent chooses action $a ^ { \dagger }$ which maximizes the sum of the threshold and the choice-specific shock, i.e., $a ^ { \dagger } = \mathrm { a r g m a x } _ { a } \{ \tilde { \pi } ( s , a ) + \epsilon _ { a } \}$ . Similarly, all non-optimal policies can be characterized by the corresponding threshold functions denoted $\pi$ . Under Assumption 2.3 the conditional choice probability $\delta$ can be explicitly expressed in terms of the respective threshold $\pi$ as (cf. Rust, 1996)
+
+$$
+\begin{array} { r } { \delta ( { a } | s ) = \exp ( \pi ( s , a ) ) \bigg / \left( \sum _ { a ^ { \prime } \in A } \exp ( \pi ( s , a ^ { \prime } ) ) \right) . } \end{array}
+$$
+
+We note that this expression induces a one-to-one mapping from the thresholds to the conditional choice probabilities. Therefore, all policies are fully characterized by their respective conditional choice probabilities. For notational simplicity, since we consider the binary action space $\mathcal { A } = \{ 0 , 1 \}$ , and the reward $r ( s , 0 )$ is normalized to 0 we denote the immediate reward $r ( s , 1 )$ as $r ( s )$ ; denote the conditional choice probability $\delta ( 0 | s )$ as $\delta ( s )$ ; and denote $\pi ( s , 1 )$ as $\pi ( s )$ .
+
+In the subsequent discussion given that the characterization of policy $\sigma$ via its threshold is equivalent to its characterization by conditional choice probability $\delta$ , we interchangeably refer to $\delta$ as the “policy." Then we rewrite the Bellman equation for a given policy $\delta$ as
+
+$$
+\begin{array} { c } { { V _ { \delta } ( s ) = \left( 1 - \delta ( s ) \right) r ( s ) - \delta ( s ) \log { ( \delta ( s ) ) } } } \\ { { - \left( 1 - \delta ( s ) \right) \log ( 1 - \delta ( s ) ) + \beta \mathbf { E } _ { \epsilon , s ^ { \prime } } \Big [ V _ { \delta } ( s ) ( s ^ { \prime } ) \Big | s \Big ] } } \end{array}
+$$
+
+Now we make two additional assumptions that are compatible with standard assumptions in the Econometrics literature.
+
+Assumption 2.4. For all states $s \in S _ { i }$ , the conditional distribution of the next period Markov state $p ( \cdot | s , 1 )$ first-order stochastically dominates distribution $p ( \cdot | s , 0 )$ , i.e., for all $\hat { s } \in \ S$ , $\mathbf { P r } _ { s ^ { \prime } } [ s ^ { \prime } \leq \hat { s } | s , 1 ] \leq \mathbf { P r } _ { s ^ { \prime } } [ s ^ { \prime } \leq \hat { s } | s , 0 ]$ .
+
+Assumption 2.5. Under the optimal policy $\tilde { \delta }$ , the value function is non-decreasing in states, i.e., $V _ { \tilde { \delta } } ( s ) \bar { \leq } V _ { \tilde { \delta } } ( s ^ { \prime } )$ for all $s , s ^ { \prime } \in S$ s.t. $s < s ^ { \prime }$ .
+
+Consider a myopic policy $\bar { \delta } ( s ) = ( \exp ( r ( s ) ) + 1 ) ^ { - 1 }$ which uses threshold $\bar { \pi } ( s ) = r ( s )$ . This policy corresponds to agent optimizing the immediate reward without considering how current actions impact future rewards. Under Assumption 2.4 and Assumption 2.5, the threshold for optimal policy is at least the threshold of myopic policy, i.e., $\tilde { \pi } ( s ) \geq \bar { \pi } ( s )$ . Hence, Lemma 2.1 holds.
+
+Lemma 2.1. The optimal policy $\tilde { \delta }$ chooses action $O$ with weakly lower probability than the myopic policy $\bar { \delta }$ in all states $s \in S$ , i.e., $\tilde { \delta } ( s ) \leq \bar { \delta } ( s )$ .
+
+# 2.3 MDP IN ECONOMICS AND POLICY GRADIENT
+
+Our motivation in this paper comes from empirical work in Economics and Marketing where optimizing agents are consumers or small firms who make dynamic decisions while observing the current state $s$ and the reward $r ( s , a )$ for their choice $a$ . These agents often have limited computational power making it difficult for them to solve the Bellman equation to find the optimal policy. They also may have only sample access to the distribution of Markov transition which further complicates the computation of the optimal policy. In this context we contrast the value function iteration method which is based on solving the fixed point problem induced by the Bellman equation and the policy gradient method.
+
+Value function iteration In the value function iterations, e.g., discussed in Jaksch et al. (2010); Haskell et al. (2016), the exact expectation in the Bellman equation (1) is replaced by an empirical estimate and then functional iteration uses the empirical Bellman equation to find the fixed point, i.e., the optimal policy. Under certain assumptions on MDPs, one can establish convergence guarantees for the value function iterations, e.g., Jaksch et al. (2010); Haskell et al. (2016). However, to run these iterations may require significant computation power which may not be practical when optimizing agents are consumers or small firms.
+
+Policy gradient In contrast to value function iterations, policy gradient algorithm and its variations are model-free sample-based methods. At a high level, policy gradient parametrizes policies $\{ \delta _ { \theta } \} _ { \theta \in \Theta }$ by $\theta \in \Theta$ and computes the gradient of the value function with respect to the current policy $\delta _ { \theta }$ and update the policy in the direction of the gradient, i.e., $\theta  \theta + \alpha \nabla _ { \theta } V _ { \delta _ { \theta } }$ . Though the individuals considered in the Economic MDP models may not compute the exact gradient with respect to a policy due to having only sample access to the Markov transition, previous work has provided approaches to produce an unbiased estimator of the gradient. For example, REINFORCE (Williams, 1992) updates the policy by $\theta  \theta + \alpha R \nabla _ { \theta } \log ( \delta _ { \theta } \bar { ( } a | s ) )$ where $R$ is the long-term payoff on path. Notice that this updating rule is simple comparing with value function iteration. The caveat of the policy gradient approach is the lack of its global convergence guarantee for a generic MDP. In this paper we show that such guarantee can be provided for the specific class of MDPs that we consider.
+
+# 3 WARM-UP: LOCAL CONCAVITY OF THE VALUE FUNCTION AT THE OPTIMAL POLICY
+
+To understand the convergence of the policy gradient, in this section we introduce our main technique and show that the concavity of the value function with respect to policies is satisfied in a fixed neighborhood around the optimal policy. We rely on the special structure of the value function induced by random shocks $\epsilon$ which essentially “smooth it" making it differentiable. We then use Bellman equation (7) to compute strong Fréchet functional derivatives of the value functions and argue that the respective second derivative is negative at the optimal policy. We use this approach in Section 4 to show the global concavity of the value function with respect to policies.
+
+By $\Delta$ we denote the convex compact set that contains all continuous functions $\delta : { \cal S }  [ 0 , 1 ]$ such that $\begin{array} { r } { 0 \le \delta ( \cdot ) \le \bar { \delta } ( \cdot ) } \end{array}$ . The Bellman equation (7) defines the functional $V _ { \delta } ( \cdot )$ . Recall that Fréchet derivative of functional $\overset { \cdot } { V _ { \delta } } ( \cdot )$ , which maps bounded linear space $\Delta$ into the space of all continuous bounded functions of $s$ , at a given $\delta ( \cdot )$ is a bounded linear functional $\mathrm Ḋ V Ḍ _ { \delta } ( \cdot )$ such that for all continuous $h ( \cdot )$ with $\| h \| _ { 2 } \leq \bar { H } \colon V _ { \delta + h } ( \cdot ) - V _ { \delta } ( \cdot ) = \mathrm Ḋ V Ḍ _ { \delta } ( \cdot ) h ( \cdot ) + o ( \| h \| _ { 2 } )$ . When functional $\mathrm Ḋ V Ḍ _ { \delta } ( \cdot )$ is also Fréchet differentiable, we refer to its Fréchet derivative as the second Fréchet derivative of functional $V _ { \delta } ( \cdot )$ and denote it $\mathrm { D } ^ { 2 } V _ { \delta } ( \cdot )$ .
+
+Theorem 3.1. Value function $V _ { \delta }$ is twice Freéchet differentiable with respect to $\delta$ at the choice probability $\tilde { \delta }$ corresponding to optimal policy and its Fréchet derivative is negative at $\tilde { \delta }$ in all states $s$ , i.e., $\mathrm { D } ^ { 2 } V _ { \tilde { \delta } } ( s ) \le 0$ .
+
+We sketch the proof idea of Theorem 3.1 and defer its formal proof to Appendix A. Start with the Bellman equation (7) of the value function, the Fréchet derivative of the value function is the fixed point of the following Bellman equation
+
+$$
+\begin{array} { r l } & { \mathrm { D } V _ { \delta } ( s ) = ( \log ( 1 - \delta ( s ) ) - \log ( \delta ( s ) ) - r ( s ) ) } \\ & { \qquad + \beta \left( \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 0 ] - \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 1 ] \right) + \beta \mathbf { E } _ { \epsilon , s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s ] , } \end{array}
+$$
+
+and
+
+$$
+\begin{array} { l } { \displaystyle \mathrm { D } ^ { 2 } V _ { \delta } ( s ) = - \frac { 1 } { \delta ( s ) ( 1 - \delta ( s ) ) } } \\ { \displaystyle \qquad - 2 \beta ( { \bf E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 1 ] - { \bf E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 0 ] ) + \beta { \bf E } _ { s ^ { \prime } } \big [ \mathrm { D } ^ { 2 } V _ { \delta } ( s ^ { \prime } ) | s \big ] . } \end{array}
+$$
+
+A necessary condition for its optimum yielding $\tilde { \delta }$ is $\mathrm Ḋ V Ḍ _ { \widetilde { \delta } } ( s ) = 0$ for all states $s$ . As a result, equation (9) implies that its second Fréchet derivative is negative for all states, i.e. $\mathrm { J } ^ { 2 } V _ { \tilde { \delta } } ( s ) \le 0$ .
+
+The Bellman equation (9) of the second Fréchet derivative suggests that $\mathrm { D } ^ { 2 } V _ { \delta } ( s ) \le 0$ for all states $s$ if
+
+$$
+\frac 1 { \delta ( s ) ( 1 - \delta ( s ) ) } + 2 \beta ( \mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 1 ] - \mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 0 ] ) \ge 0
+$$
+
+The first term in the inequality (5) is always positive for all policies in $\Delta$ , but the second term can be arbitrary small. In the next section, we will introduce a nature smoothness assumption on MDP (i.e., Lipschitz MDP) and show that the local concavity can be extended to global concavity, which implies that the policy gradient algorithm for our problem converges globally under this assumption.
+
+# 4 GLOBAL CONCAVITY OF THE VALUE FUNCTION
+
+In this section, we introduce the notion of the Lipschitz Markov decision process, and Lipschitz policy space. We then restrict our attention to this subclass of MDPs. Our main result shows the optimal policy belongs to the Lipschitz policy space and the policy gradient globally converges in that space. We defer all the proofs of the results in this section to Appendix B.
+
+# 4.1 LIPSCHITZ MARKOV DECISION PROCESS
+
+Lipschitz Markov decision process has the property that for two state-action pairs that are close with respect to Euclidean metric in $s$ , their immediate rewards $r$ and Markovian transition $\mathcal { P }$ should be close with respect to the Kantorovich or $L _ { 1 }$ -Wasserstein metric. Kantorovich metric is, arguable, the most common metric used used in the analysis of MDPs (cf. Hinderer, 2005; Rachelson and Lagoudakis, 2010; Pirotta et al., 2015).
+
+Definition 4.1 (Kantorovich metric). For any two probability measures $p , q ,$ , the Kantorovich metric between them is
+
+$$
+{ \mathcal { K } } ( p , q ) = \operatorname* { s u p } _ { f } \left\{ \left| \int _ { X } f d ( p - q ) \right| : f i s \ I - L i p s c h i t z \ c o n t i n u o u s \right\}
+$$
+
+Definition 4.2 (Lipschitz MDP). A Markov decision process is $( L _ { r } , L _ { p } )$ -Lipschitz $i f$
+
+$$
+\begin{array} { r l r l } & { \forall s , s ^ { \prime } \in S } & & { | r ( s ) - r ( s ^ { \prime } ) | \le L _ { r } | s - s ^ { \prime } | } \\ & { \forall s , s ^ { \prime } \in S , a , a ^ { \prime } \in \mathcal { A } } & & { K ( p ( \cdot | s , a ) , p ( \cdot | s ^ { \prime } , a ^ { \prime } ) ) \le L _ { p } \left( | s - s ^ { \prime } | + | a - a ^ { \prime } | \right) } \end{array}
+$$
+
+# 4.2 CHARACTERIZATION OF THE OPTIMAL POLICY
+
+Our result in Section 3, demonstrates that the second Fréchet derivative of $V _ { \delta }$ with respect to $\delta$ is negative for a given policy $\delta$ when inequality (5) holds. To bound the second term of (5) from below, i.e., $\mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( \bar { s ^ { \prime } } ) | s , \bar { 0 } ] - \dot { \mathbf { E } } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | \bar { s } , 1 ] ,$ , it is sufficient to show that Fréchet derivative $\mathrm Ḋ V Ḍ _ { \delta } ( \cdot )$ is Lipschitz-continuous. Even though we already assume that the Markov transition is Lipschitz, it is still possible that $\mathrm Ḋ V Ḍ _ { \delta }$ is not Lipschitz: Bellman equation (8) for $\mathrm Ḋ V Ḍ _ { \delta }$ depends on policy $\delta ( s )$ via $\log ( \bar { 1 } - \delta ( s ) ) - \log ( \delta ( s ) )$ , which can be non-Lipschitz in state $s$ for general policies $\delta$ . Therefore, to guarantee Lipschitzness of the Fréchet derivative of the value function it is necessary to restrict attention to the space of Lipschitz policies. In this subsection, we show that this restriction is meaningful since the optimal policy is Lipschitz.
+
+Theorem 4.1. Given $( L _ { r } , L _ { p } )$ -Lipschitz MDP, the optimal policy $\tilde { \delta }$ satisfies
+
+$$
+\left| \log \left( \frac { 1 - \tilde { \delta } ( s ) } { \tilde { \delta } ( s ) } \right) - \log \left( \frac { 1 - \tilde { \delta } ( s ^ { \dagger } ) } { \tilde { \delta } ( s ^ { \dagger } ) } \right) \right| \leq \left( L _ { r } + \frac { 2 \beta R _ { \operatorname* { m a x } } L _ { p } } { 1 - \beta } \right) \left| s - s ^ { \dagger } \right|
+$$
+
+for all state $s , s ^ { \dagger } \in { \mathcal { S } }$ where $R _ { \mathrm { m a x } } = \operatorname* { m a x } _ { s \in \mathcal { S } } r ( s )$ is the maximum of the immediate reward $r$ over $s$ .
+
+# 4.3 CONCAVITY OF THE VALUE FUNCTION WITH RESPECT TO LIPSCHITZ POLICIES
+
+In this subsection, we present our main result showing the global concavity of the value function for our specific class of Lipschitz MDPs with unobserved heterogeneity over the space of Lipschitz policies.
+
+Definition 4.3. Given $( L _ { r } , L _ { p } )$ -Lipschitz MDP, define its Lipschitz policy space $\Delta$ as
+
+$\Delta = \{ \delta : \delta ( s ) \leq \bar { \delta } ( s ) \forall s \in \mathcal { S }$ and
+
+$$
+| \log ( \frac { 1 - \delta ( s ) } { \delta ( s ) } ) - \log ( \frac { 1 - \delta ( s ^ { \dagger } ) } { \delta ( s ^ { \dagger } ) } ) | \leq ( L _ { r } + \frac { 2 \beta R _ { \operatorname* { m a x } } L _ { p } } { 1 - \beta } ) | s - s ^ { \dagger } | \forall s , s ^ { \dagger } \in \mathcal { S } \} ,
+$$
+
+where $\bar { \delta }$ is the myopic policy.
+
+Theorem 4.1 and Lemma 2.1 imply that the optimal policy $\tilde { \delta }$ lies in this Lipschitz policy space $\Delta$ for any Lipschitz MDP.
+
+Definition 4.4 (Condition for global convergence). We say that $( L _ { r } , L _ { p } )$ -Lipschitz MDP satisfies the sufficient condition for global convergence $i f$
+
+$$
+2 \beta L _ { p } < 1 a n d \frac { 2 \beta L _ { p } } { 1 - 2 \beta L _ { p } } \left( 2 L _ { r } + \frac { 4 \beta R _ { \mathrm { m a x } } L _ { p } } { 1 - \beta } \right) \leq \frac { \left( \exp ( R _ { \mathrm { m i n } } ) + 1 \right) ^ { 2 } } { \exp ( R _ { \mathrm { m i n } } ) } .
+$$
+
+Theorem 4.2. Given $( L _ { r } , L _ { p } )$ -Lipschitz MDP which satisfies the condition for global convergence (6), value function $V _ { \delta }$ is concave with respect to policy $\delta$ in the Lipschitz policy space $\Delta$ , i.e., $\mathrm { D } ^ { 2 } V _ { \delta } ( s ) \dot { \le } 0$ for all $s \in \mathcal S , \delta \in \Delta$ .
+
+# 4.4 THE RATE OF GLOBAL CONVERGENCE OF THE POLICY GRADIENT ALGORITHM
+
+In this subsection, we establish the rate of global convergence a simple version of the policy gradient algorithm assuming oracle access to the Fréchet derivative of the value function. While this analysis provides only a theoretical guarantee, as discussed in Section 2.3, in practice the individuals are able to produce an unbiased estimator of the exact gradient. As a result, the practical application of the policy gradient algorithm would only need to adjust for the impact of stochastic noise in the estimator.
+
+Since we assume that individuals know the immediate reward function $r$ , the algorithm can be initialized at the myopic policy $\bar { \delta }$ with threshold $\bar { \pi } ( s ) = r ( s )$ , which is in the Lipschitz policy space $\Delta$ . From Lemma 2.1 it follows that the myopic policy is pointwise in $s$ greater than the optimal policy, i.e., $\bar { \delta } ( s ) \leq \tilde { \delta } ( s )$ . Consider policy $\underline { { \delta } }$ with threshold $\begin{array} { r } { \underline { { \pi } } ( s ) = r ( s ) + \frac { \bar { \beta } } { 1 - \beta } R _ { \mathrm { m a x } } - \frac { \beta } { 2 } R _ { \mathrm { m i n } } } \end{array}$ . Note that Bellman equation (7) implies that $V ( s )$ is between $\frac { R _ { \mathrm { m i n } } } { 2 }$ and Rmax1−β for all states s. Thus, policy $\underline { { \delta } }$ pointwise bounds the optimal policy $\tilde { \delta }$ from below, i.e., $\underline { { \delta } } ( s ) \leq \widetilde { \delta } ( s )$ . Our convergence rate result applies to the policy gradient within the bounded Lipschitz policy set $\hat { \Delta }$ .
+
+Definition 4.5. Given $( L _ { r } , L _ { p } )$ -Lipschitz MDP, define its bounded Lipschitz policy space $\hat { \Delta }$ as $\hat { \Delta } = \{ \delta : \underline { { { \delta } } } ( s ) \leq \delta ( s ) \leq \bar { \delta } ( s ) \forall s \in \mathcal { S }$ and
+
+$$
+| \log ( \frac { 1 - \delta ( s ) } { \delta ( s ) } ) - \log ( \frac { 1 - \delta ( s ^ { \dagger } ) } { \delta ( s ^ { \dagger } ) } ) | \leq ( L _ { r } + \frac { 2 \beta R _ { \operatorname* { m a x } } L _ { p } } { 1 - \beta } ) | s - s ^ { \dagger } | \forall s , s ^ { \dagger } \in \mathcal { S } \} .
+$$
+
+For simplicity of notation, we introduce constants $m$ and $M$ which only depend on $\beta , R _ { \mathrm { m i n } } , R _ { \mathrm { m a x } }$ $L _ { r }$ and $L _ { p }$ , whose exact expressions are deferred to the supplementary material for this paper.
+
+Theorem 4.3. Given a $( L _ { r } , L _ { p } )$ -Lipschitz MDP, which satisfies the condition for global convergence (6) and constants m and $M$ defined above, for any step size $\begin{array} { r } { \alpha \leq \frac { 1 } { M } } \end{array}$ , the policy gradient initialized at the myopic policy $\bar { \delta }$ and updating as $\delta \gets \alpha \nabla _ { \delta } V _ { \delta }$ in the bounded Lipschitz policy space $\hat { \Delta }$ after $k$ iterations, it produces policy $\delta ^ { ( k ) }$ satisfying
+
+$$
+V _ { \tilde { \delta } } ( s ) - V _ { \delta ^ { ( k ) } } ( s ) \leq \frac { ( 1 - \alpha m ) ^ { k } } { \left( \exp ( R _ { \operatorname* { m i n } } ) + 1 \right) ^ { 2 } } \qquad a t a l l s \in \mathcal { S } .
+$$
+
+# 5 EMPIRICAL APPLICATION
+
+To demonstrate the performance of the algorithm, we use the data from Rust (1987) which made the standard benchmark for the Econometric analysis of MDPs. The paper estimates the cost associated with maintaining and replacing bus engines using data from maintenance records from Madison Metropolitan Bus City Company over the course of 10 years (December, 1974—May, 1985). The data contains monthly observations on the mileage of each bus as well as the dates of major maintenance events (such as bus engine replacement).
+
+Rust (1987) assumes that the engine replacement decisions follow an optimal stopping policy derived from solving a dynamic discrete choice model of the type that we described earlier. Using this assumption and the data, he estimates the cost of operating a bus as a function of the running mileage as well as the cost of replacing the bus engine. We use his estimates of the parameters of the return function and the state transition probabilities (bus mileage) to demonstrate convergence of the gradient descent algorithm.
+
+In Rust (1987) the state $s _ { t }$ is the running total mileage of the bus accumulated by the end of period $t$ The immediate reward is specified as a function of the running mileage as:
+
+$$
+r ( s _ { t } , a , \theta _ { 1 } ) = { \left\{ \begin{array} { l l } { - \mathrm { R C } + \epsilon _ { t 1 } , } & { { \mathrm { i f ~ } } a = 1 } \\ { - c ( s _ { t } , \theta _ { 1 } ) + \epsilon _ { t 0 } , } & { { \mathrm { i f ~ } } a = 0 } \end{array} \right. }
+$$
+
+where RC is the cost of replacing the engine, $c ( s _ { t } , \theta _ { 1 } )$ is the cost of operating a bus that has $s _ { t }$ miles.
+
+Following Rust (1987), we take $c ( s _ { t } , \theta _ { 1 } ) = \theta _ { 1 } s _ { t }$ . Further, as in the original paper, we discretize the mileage taking values in the range from 0 to 175 miles into an even grid of 2,571 intervals. Given the observed monthly mileage, Rust (1987) assumes that transitions on the grid can only be of increments $0 , 1 , 2 , 3$ and 4. Therefore, transition process for discretized mileage is fully specified by just four parameters $\theta _ { 2 j } = \mathbf { P r } [ s _ { t + 1 } = s _ { t } + j | s _ { t } , a = 0 ]$ , $j = 0 , 1 , 2 , 3$ . Table 1 describes parameter values that we use directly from Rust (1987).
+
+Table 1: parameter values in from Rust (1987).   
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>RC</td><td>11.7257</td></tr><tr><td>01</td><td>0.001× 2.45569</td></tr><tr><td>(020,021,022,023)</td><td>(0.0937, 0.4475, 0.4459, 0.0127)</td></tr><tr><td>β</td><td>0.99</td></tr></table>
+
+We use the gradient descent algorithm to update the policy threshold $\pi : \epsilon _ { 1 } + \pi \geq \epsilon _ { 0 } \Rightarrow a = 1$ , where $a = 1$ denotes the decision to replace the engine. We set the learning rate using the RMSprop method1.
+
+We use “the lazy projection" method to guarantee the search over Lipschitz policy space. The policy space is parametrized by the vector of thresholds $( \pi _ { 1 } , \ldots , \pi _ { N } )$ corresponding to discretized state space $( s _ { 1 } , \dotsc , s _ { N } )$ . It is initialized at the myopic policy, i.e. $\pi _ { 1 } ^ { ( 0 ) } =$ $u ( s _ { 1 } ) , \ldots , \pi _ { N } ^ { ( 0 ) } = u ( s _ { N } )$ . At step $k$ the algorithm updates the thresholds to the value $\pi _ { i } ^ { ( k * ) } =$ $\pi _ { i } ^ { ( k - 1 ) } - \alpha \mathrm { D } _ { \delta ^ { ( k - 1 ) } } V ( s _ { i } ) \mathcal { L } ( \pi _ { i } ^ { ( k - 1 ) } ) ( 1 - \mathcal { L } ( \pi _ { i } ^ { ( k - 1 ) } ) )$ $\delta _ { j } ^ { ( k ) } = \mathcal { L } ( \pi _ { j } ^ { ( k - 1 ) } )$ ) for $i , j = 1 , \dots , N$ i . To make the“lazy projection" updated values , where is the logistic function and policy $\pi _ { i } ^ { ( k * ) }$ are adjusted to the closest monotone set of values $\pi _ { 1 } ^ { ( k ) } \leq \pi _ { 2 } ^ { ( k ) } \leq \ldots \leq \pi _ { N } ^ { ( k ) }$ . The algorithm terminates at step $k$ where the norm $\operatorname* { m a x } _ { i } | \mathrm { D } V _ { \delta ^ { ( k ) } } ( s _ { i } ) | \leq \tau$ for a given tolerance $\tau$ .2 The formal definition of lazy projection can be found in Appendix C.
+
+Figure 3 demonstrates convergence properties of our considered version of the policy gradient algorithm. We used the “oracle" versions of the gradient and the value function that were obtained by solving the corresponding Bellman equations. We initialized the algorithm using the myopic threshold $\bar { \pi } ( s ) \bar { = } - \mathbf { R C } + c \bar { ( } s , \theta _ { 1 } )$ ; with the convergence criterion set to be based on the value maxi $| \mathrm { D } V _ { \delta } ( s _ { i } ) | ^ { 3 }$
+
+In the original model in Rust (1987), the discount factor used when estimating parameters of the cost function was very close to 1. However, performance of the algorithm improves drastically when the discount factor is reduced. This feature is closely related to the Hadamard stability of the solution of the Bellman equation (e.g. observed in Bajari et al. (2013)) and is not algorithm-specific. In all of the follow-up analysis by the same author (e.g. Rust (1996)) the discount factor is set to more moderate values of .99 or .9 indicating that these performance issues were indeed observed with the settings in Rust (1987). Figure 3 illustrates the performance of the algorithm for the case where the discount factor is set to $0 . { \overset { - } { 9 } } 9 ^ { 4 }$ . For the same convergence criterion, the algorithm converges much faster.
+
+![](images/66399ded3a2e90e744f558e5018f688a4e600a7648a7420a3f4dcdf860a15f62.jpg)  
+Figure 1: Convergence of gradient descent, discount factor $\beta = 0 . 9 9$
+
+![](images/3d6f140ded77923f326665df3e8a2d54d4ea59be59e33cf6837a5673cb233141.jpg)  
+Figure 2: Performance of the norm $\mathrm { m a x } _ { i }$ $\left| \mathrm Ḋ V Ḍ _ { \delta } ( s _ { i } ) \right|$ and the second derivative $\operatorname { n a x } _ { i } \left| \operatorname { D } ^ { 2 } V _ { \delta } ( s _ { i } ) \right|$ discount factor $\beta = 0 . 9 9$
+
+REFERENCES   
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+Dunford, N. and Schwartz, J. T. (1957). Linear Operators. Part 1: General Theory. New York Interscience.   
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+Hotz, V. J. and Miller, R. A. (1993). Conditional choice probabilities and the estimation of dynamic models. The Review of Economic Studies, 60(3):497–529.   
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+Nekipelov, D., Syrgkanis, V., and Tardos, E. (2015). Econometrics for learning agents. In Proceedings of the Sixteenth ACM Conference on Economics and Computation, pages 1–18. ACM.   
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+Williams, R. J. (1992). Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256.
+
+# APPENDIX
+
+# A OMITTED PROOF FOR THEOREM 3.1
+
+Theorem 3.1. Value function $V _ { \delta }$ is twice Freéchet differentiable with respect to $\delta$ at the choice probability $\tilde { \delta }$ corresponding to optimal policy and its Fréchet derivative is negative at $\tilde { \delta }$ in all states $s$ , i.e., $\mathrm { D } ^ { 2 } V _ { \widetilde { \delta } } ( s ) \le 0$ .
+
+Proof. We start with the Bellman equation of the value function.
+
+$$
+\begin{array} { c } { { V _ { \delta } ( s ) = \left( 1 - \delta ( s ) \right) r ( s ) - \delta ( s ) \log { ( \delta ( s ) ) } } } \\ { { - \left( 1 - \delta ( s ) \right) \log ( 1 - \delta ( s ) ) + \beta \mathbf { E } _ { \epsilon , s ^ { \prime } } \Big [ V _ { \delta } ( s ) ( s ^ { \prime } ) \Big | s \Big ] } } \end{array}
+$$
+
+First of all, note that in (7) the first three terms on the right hand side of the equation simple nonlinear functions $\delta ( \cdot )$ and thus the directional derivative with respect to $\delta ( \cdot )$ can be taken as an ordinary derivative with respect to $\delta$ as a parameter. Next note that if functional ${ \dot { J } } _ { \delta } ( \cdot )$ is directionally differentiable with respect to $\delta$ and for all $h ( \cdot )$ , $\begin{array} { r } { \frac { d } { d \tau } J _ { \delta + \tau h } ( \cdot ) | _ { \tau = 0 } / h ( \cdot ) } \end{array}$ is invariant, then $J _ { \delta } ( \cdot )$ is Fréchet differentiable with respect to $\delta$ and the obove ratio is its Fréchet derivative. As a result, the Fréchet derivative of simple functional $\left( 1 - \delta ( s ) \right) r ( s ) - \delta ( s ) \log \left( \delta ( s ) \right) - ( 1 - \delta ( s ) ) \log ( 1 - \delta ( s ) )$ with respect to $\delta ( \cdot )$ exists and equal to $\log ( 1 - \delta ( s ) ) - \log ( \delta ( s ) ) - r ( s )$ . This expression is itself a Freéchet-differentiable functional with Fréchet derivative equal to $- \mathrm { i } / ( \delta ( s ) ( 1 - \delta ( s ) ) ) .$ , meaning that the original functional $\begin{array} { r } { \left( 1 - \delta ( s ) \right) r ( s ) - \delta ( s ) \log \left( \delta ( s ) \right) - ( 1 - \delta ( s ) ) \log ( 1 - \delta ( s ) ) } \end{array}$ is twice Fréchet differentiable with the second Fréchet derivative $- 1 / ( \delta ( s ) ( 1 - \delta ( s ) ) )$ . Whenever the state transition is affected by the individual decision we need to consider decomposition of the conditional expectation with respect to the future state:
+
+$$
+\begin{array} { r } { \mathbf { E } _ { \epsilon , s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s ] = \left( 1 - \delta ( s ) \right) \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 1 ] + \delta ( s ) \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 0 ] . } \end{array}
+$$
+
+er standard technical conditions that allow the swap of the derivative and the integral
+
+$$
+\mathrm { D } \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s ] = ( \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 0 ] - \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 1 ] ) + \mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s ]
+$$
+
+Thus, the Fréchet derivative of the value function should be the fixed point of the following Bellman equation
+
+$$
+\begin{array} { r l } & { \mathrm { D } V _ { \delta } ( s ) = ( \log ( 1 - \delta ( s ) ) - \log ( \delta ( s ) ) - r ( s ) ) } \\ & { \qquad + \beta \left( \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 0 ] - \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 1 ] \right) + \beta \mathbf { E } _ { \epsilon , s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s ] , } \end{array}
+$$
+
+and
+
+$$
+\begin{array} { l } { \displaystyle \mathrm { D } ^ { 2 } V _ { \delta } ( s ) = - \frac { 1 } { \delta ( s ) ( 1 - \delta ( s ) ) } } \\ { \displaystyle \qquad - 2 \beta ( { \bf E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 1 ] - { \bf E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 0 ] ) + \beta { \bf E } _ { s ^ { \prime } } [ \mathrm { D } ^ { 2 } V _ { \delta } ( s ^ { \prime } ) | s ] . } \end{array}
+$$
+
+Given that both these equations are Type II Fredholm integral equations for $\mathrm Ḋ V Ḍ _ { \delta } ( \cdot )$ and $\mathrm { D } ^ { 2 } V _ { \delta } ( \cdot )$ which have unique solutions whenever $\beta < 1$ that are bounded and continuous (see Dunford and Schwartz (1957)) and, thus, unique solutions for both equations exist and $V _ { \delta } ( \cdot )$ is indeed Fréchetdifferentiable. This means that the necessary condition for its optimum yielding $\tilde { \delta }$ is $\mathrm Ḋ V Ḍ _ { \tilde { \delta } } ( s ) = 0$ for all states $s$ . As a result, equation (9) implies that its second Fréchet derivative is negative for all states, i.e., $\mathrm { D } ^ { 2 } V _ { \tilde { \delta } } ( s ) \le 0$ . □
+
+# B OMITTED PROOFS IN SECTION 4
+
+B.1 OMITTED PROOF OF THEOREM 4.1
+
+Theorem 4.1. Given an $( L _ { r } , L _ { p } )$ -Lipschitz MDP, the optimal policy $\tilde { \delta }$ satisfies
+
+$$
+\left| \log \left( \frac { 1 - \tilde { \delta } ( s ) } { \tilde { \delta } ( s ) } \right) - \log \left( \frac { 1 - \tilde { \delta } ( s ^ { \dagger } ) } { \tilde { \delta } ( s ^ { \dagger } ) } \right) \right| \leq \left( L _ { r } + \frac { 2 \beta R _ { \operatorname* { m a x } } L _ { p } } { 1 - \beta } \right) \left| s - s ^ { \dagger } \right|
+$$
+
+for all state $s , s ^ { \dagger } \in { \mathcal { S } }$ where $R _ { \mathrm { m a x } } = \operatorname* { m a x } _ { s \in \mathcal { S } } r ( s )$ is the maximum of the immediate reward $r$ over $s$ .
+
+Proof. At the optimal policy $\tilde { \delta }$ , the Fréchet derivative of the value function is zero, i.e., $\mathrm Ḋ V Ḍ _ { \widetilde { \delta } } ( s ) = 0$ for all state $s$ . Therefore, from the Bellman equation (8) we establish that
+
+$$
+\log \left( \frac { 1 - \tilde { \delta } ( s ) } { \tilde { \delta } ( s ) } \right) = r ( s ) + \beta \left( \mathbf { E } _ { s ^ { \prime } } \left[ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s , 1 \right] - \mathbf { E } _ { s ^ { \prime } } \left[ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s , 0 \right] \right)
+$$
+
+Thus, for all states $s , s ^ { \dagger } \in { \mathcal { S } }$ ,
+
+$$
+\begin{array} { r l } & { \quad \left| \mathbf { E } _ { s ^ { \prime } } [ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s , a ] - \mathbf { E } _ { s ^ { \prime } } [ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s ^ { \dagger } , a ] \right| } \\ & { = \left| \int _ { s ^ { \prime } \in S } V _ { \tilde { \delta } } ( s ^ { \prime } ) ( p ( s ^ { \prime } | s , a ) - p ( s ^ { \prime } | s ^ { \dagger } , a ) ) d s ^ { \prime } \right| } \\ & { = \frac { R _ { \operatorname* { m a x } } } { 1 - \beta } \left| \int _ { s ^ { \prime } \in S } \frac { ( 1 - \beta ) } { R _ { \operatorname* { m a x } } } V _ { \tilde { \delta } } ( s ^ { \prime } ) ( p ( s ^ { \prime } | s , a ) - p ( s ^ { \prime } | s ^ { \dagger } , a ) ) d s ^ { \prime } \right| } \\ & { \leq \frac { R _ { \operatorname* { m a x } } } { 1 - \beta } \underset { \| \boldsymbol { f } \| _ { L ^ { \infty } } } { \operatorname* { s u p } } \left\{ \left| \int _ { s ^ { \prime } \in S } f ( s ^ { \prime } ) ( p ( s ^ { \prime } | s , a ) - p ( s ^ { \prime } | s ^ { \dagger } , a ) ) d s ^ { \prime } \right| \right\} } \\ & { = \frac { R _ { \operatorname* { m a x } } } { 1 - \beta } K ( p ( \cdot | s , a ) , p ( \cdot | s ^ { \dagger } , a ) ) \leq \frac { R _ { \operatorname* { m a x } } L _ { p } } { 1 - \beta } | s - s ^ { \dagger } | } \end{array}
+$$
+
+where we use upper bounds $\begin{array} { r } { \operatorname* { s u p } _ { s \in \mathcal { S } } V _ { \tilde { \delta } } ( s ) \le \frac { R _ { \operatorname* { m a x } } } { 1 - \beta } } \end{array}$ and k (1−β)R Vδ˜(s0)kL ≤ 1. Thus,
+
+$$
+\begin{array} { r l } & { \left| \log \left( \frac { 1 - \tilde { \delta } ( s ) } { \tilde { \delta } ( s ) } \right) - \log \left( \frac { 1 - \tilde { \delta } ( s ^ { \prime } ) } { \tilde { \delta } ( s ^ { \prime } ) } \right) \right| } \\ & { = | r ( s ) - r ( s ^ { \prime } ) + \beta \left( \mathbf { E } _ { s ^ { \prime } } \left[ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s , 1 \right] - \mathbf { E } _ { s ^ { \prime } } \left[ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s , 0 \right] \right) } \\ & { \qquad - \beta \left( \mathbf { E } _ { s ^ { \prime } } \left[ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s ^ { \dagger } , 1 \right] - \mathbf { E } _ { s ^ { \prime } } \left[ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s ^ { \dagger } , 0 \right] \right) | } \\ & { \leq \left| r ( s ) - r ( s ^ { \dagger } ) \right| + \beta \left| \mathbf { E } _ { s ^ { \prime } } \left[ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s , 1 \right] - \mathbf { E } _ { s ^ { \prime } } \left[ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s ^ { \dagger } , 1 \right] \right| } \\ & { \qquad + \beta \left| \mathbf { E } _ { s ^ { \prime } } \left[ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s , 0 \right] - \mathbf { E } _ { s ^ { \prime } } \left[ V _ { \tilde { \delta } } ( s ^ { \prime } ) | s ^ { \dagger } , 0 \right] \right| } \\ & { \leq \left( L _ { r } + \frac { 2 \beta R _ { \operatorname* { m a x } } L _ { p } } { 1 - \beta } \right) \left| s - s ^ { \dagger } \right| } \end{array}
+$$
+
+# B.2 OMITTED PROOF OF THEOREM 4.2
+
+Theorem 4.2. Given an $( L _ { r } , L _ { p } )$ -Lipschitz MDP which satisfies the condition for global convergence (6), the value function $V _ { \delta }$ is concave with respect to policy $\delta$ in the Lipschitz policy space $\Delta$ , i.e., $\mathrm { D } ^ { 2 } V _ { \delta } ( s ) \le 0$ for all $s \in \mathcal S , \delta \in \Delta$ .
+
+To show Theorem 4.2, we first introduce the following lemma establishing Lipschitz continuity of the Fréchet derivative of the value function.
+
+Lemma B.1. Given a $( L _ { r } , L _ { p } )$ -Lipschitz MDP, for all policies $\delta$ in the the Lipschitz policy space $\Delta$ , the Fréchet derivative of the respective value function $\mathrm Ḋ V Ḍ _ { \delta } ( \cdot )$ is $\left( \frac { 2 L _ { r } + \frac { 4 \beta R _ { \mathrm { m a x } } L _ { p } } { 1 - \beta } } { 1 - 2 \beta L _ { p } } \right)$ -Lipschitz
+
+continuous, i.e., for all states $s , s ^ { \dagger } \in { \mathcal { S } }$ ,
+
+$$
+\bigl | \mathrm { D } V _ { \delta } ( s ) - \mathrm { D } V _ { \delta } ( s ^ { \dagger } ) \bigr | \leq \left( \frac { 2 L _ { r } + \frac { 4 \beta R _ { \mathrm { m a x } } L _ { p } } { 1 - \beta } } { 1 - 2 \beta L _ { p } } \right) \bigl | s - s ^ { \dagger } \bigr | .
+$$
+
+Proof. We begin with the Bellman equation (8) for the Fréchet derivative of value function.
+
+$$
+\begin{array} { r l } & { \mathrm { D } V _ { \delta } ( s ) = \log \left( \frac { 1 - \delta ( s ) } { \delta ( s ) } \right) - r ( s ) } \\ & { \qquad + \left. \beta \left( \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 0 ] - \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 1 ] \right) + \beta \mathbf { E } _ { \epsilon , s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s ] \right) } \end{array}
+$$
+
+We use the concept of the contraction mapping to prove the result of the Lemma.
+
+Definition B.1. Let $T : X  X$ be a mapping from a metric space $X$ to itself,
+
+• $T$ is a contraction mapping (with modulus $\gamma \in [ 0 , 1 )$ ) i $f \rho ( T ( x ) , T ( y ) ) \leq \gamma \rho ( x , y )$ for all $x , y \in X$ , where $\rho$ is a metric on $X$ .   
+• $x$ is a fixed point of $T$ if $T ( x ) = x$ .
+
+Lemma B.2. Suppose that $X$ is a complete metric space and that $T : X \to X$ is a contraction mapping with modulus $\gamma .$ . Then,
+
+• $T$ has a unique fixed point $x ^ { * }$ .   
+• If $X ^ { \prime } \subseteq X$ is a closed subset for which $T ( X ^ { \prime } ) \subseteq X ^ { \prime }$ , then $x ^ { * } \in X ^ { \prime }$ .
+
+Consider the contraction mapping $\begin{array} { r } { T _ { \delta } ( x ) ( s ) ~ = ~ \log \left( \frac { 1 - \delta ( s ) } { \delta ( s ) } \right) ~ - ~ r ( s ) ~ + ~ \beta \left( \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 0 ] ~ - ~ \right. } \end{array}$ $\mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 1 ] ) + \beta \mathbf { E } _ { \epsilon , s ^ { \prime } } [ x ( s ^ { \prime } ) | s ]$ , then the Bellman equation implies that $\mathrm Ḋ V Ḍ _ { \delta }$ is the fixed point of contraction mapping $T _ { \delta }$ . Since the Lipschitz continuity property forms a closed subset, by Lemma B.2, it is sufficient to show for any $L _ { D V }$ -Lipschitz continuous $x$ , $T _ { \delta } ( x )$ is also $L _ { D V }$ -Lipschitz continuous, where $\begin{array} { r } { L _ { D V } = \frac { 2 L _ { r } + \frac { 4 \beta R _ { \mathrm { m a x } } L _ { p } } { 1 - \beta } } { 1 - 2 \beta L _ { p } } } \end{array}$ . Thus, consider states $s , s ^ { \dagger } \in { \mathcal { S } }$ ,
+
+$$
+\begin{array} { r l } & { \quad \left| T ( x ) ( s ) - T ( x ) ( s ^ { \dagger } ) \right| } \\ & { \leq \left| \log \left( \frac { 1 - \delta ( s ) } { \delta ( s ) } \right) - \log \left( \frac { 1 - \delta ( s ^ { \dagger } ) } { \delta ( s ^ { \dagger } ) } \right) \right| + \left| r ( s ) - r ( s ^ { \dagger } ) \right| } \\ & { \qquad + \left. \beta \left| { \bf E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 0 ] - { \bf E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s ^ { \dagger } , 0 ] \right| \right. } \\ & { \qquad + \left. \beta \left| { \bf E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s , 1 ] - { \bf E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) | s ^ { \dagger } , 1 ] \right| \right. } \\ & { \qquad + \left. \beta \left| { \bf E } _ { s ^ { \prime } } [ x ( s ^ { \prime } ) | s , 0 ] \delta ( s ) - { \bf E } _ { s ^ { \prime } } [ x ( s ^ { \prime } ) | s ^ { \dagger } , 0 ] \right. \delta ( s ^ { \dagger } ) \right| } \\ & { \qquad + \left. \beta \left| { \bf E } _ { s ^ { \prime } } [ x ( s ^ { \prime } ) | s , 1 ] \left( 1 - \delta ( s ) \right) - { \bf E } _ { s ^ { \prime } } [ x ( s ^ { \prime } ) | s ^ { \dagger } , 1 ] \right. \left( 1 - \delta ( s ^ { \dagger } ) \right) \right| } \end{array}
+$$
+
+where
+
+$$
+\left| \log \left( \frac { 1 - \delta ( s ) } { \delta ( s ) } \right) - \log \left( \frac { 1 - \delta ( s ^ { \dagger } ) } { \delta ( s ^ { \dagger } ) } \right) \right| \leq \left( L _ { r } + \frac { 2 \beta R _ { \operatorname* { m a x } } L _ { p } } { 1 - \beta } \right) \left| s - s ^ { \dagger } \right|
+$$
+
+by the same calculation in the proof of Theorem 4.1, for $a = 0 , 1$ ,
+
+$$
+\beta \left| \mathbf { E } _ { s ^ { \prime } } [ V _ { \delta } ( s ^ { \prime } ) ] s , a ] - \mathbf { E } _ { s ^ { \prime } } \left[ V _ { \delta } ( s ^ { \prime } ) \vert s ^ { \dagger } , a \right] \right| \leq \left( L _ { r } + \frac { 2 \beta R _ { \operatorname* { m a x } } L _ { p } } { 1 - \beta } \right) \left| s - s ^ { \dagger } \right|
+$$
+
+and
+
+$$
+\begin{array} { r l } & { \quad \left| \mathbf { E } _ { s ^ { \prime } } [ x ( s ^ { \prime } ) | s , 0 ] \delta ( s ) - \mathbf { E } _ { s ^ { \prime } } \bigl [ x ( s ^ { \prime } ) | s ^ { \dagger } , 0 \bigr ] \delta ( s ^ { \dagger } ) \right| } \\ & { = \left| \displaystyle \int _ { s ^ { \prime } \in S } ( \delta ( s ) - \delta ( s ^ { \dagger } ) ) x ( s ^ { \prime } ) ( p ( s ^ { \prime } | s , a ) - p ( s ^ { \prime } | s ^ { \dagger } , a ) ) d s ^ { \prime } \right| } \\ & { = L _ { D V } \left| \displaystyle \int _ { s ^ { \prime } \in S } \frac { ( \delta ( s ) - \delta ( s ^ { \dagger } ) ) x ( s ^ { \prime } ) } { L _ { D V } } ( p ( s ^ { \prime } | s , a ) - p ( s ^ { \prime } | s ^ { \dagger } , a ) ) d s ^ { \prime } \right| } \\ & { \le L _ { D V } \underset { \| f \| _ { L } \leq 1 } { \operatorname* { s u p } } \left. \left| \displaystyle \int _ { s ^ { \prime } \in S } f ( s ^ { \prime } ) ( p ( s ^ { \prime } | s , a ) - p ( s ^ { \prime } | s ^ { \dagger } , a ) ) d s ^ { \prime } \right| \right. } \\ & { = L _ { D V } K ( p ( \cdot | s , a ) , p ( \cdot | s ^ { \dagger } , a ) ) \le L _ { D V } L _ { p } \left| s - s ^ { \dagger } \right| } \end{array}
+$$
+
+where we use the bound $\left| \delta ( s ) - \delta ( s ^ { \dagger } ) \right| \leq 1$ and thus $\| \frac { ( \delta ( s ) - \delta ( s ^ { \dagger } ) ) x ( s ^ { \prime } ) } { L _ { D V } } \| _ { L } \leq 1$ . Similarly,
+
+$$
+\left| \mathbf { E } _ { s ^ { \prime } } [ x ( s ^ { \prime } ) | s , 1 ] \left( 1 - \delta ( s ) \right) - \mathbf { E } _ { s ^ { \prime } } \left[ x ( s ^ { \prime } ) | s ^ { \dagger } , 1 \right] \left( 1 - \delta ( s ^ { \dagger } ) \right) \right| \leq L _ { D V } L _ { p } \left| s - s ^ { \dagger } \right|
+$$
+
+Combining all the bounds, we obtain that
+
+$$
+\left| T ( x ) ( s ) - T ( x ) ( s ^ { \dagger } ) \right| \le \left( 2 L _ { r } + \frac { 4 \beta R _ { \mathrm { m a x } } L _ { p } } { 1 - \beta } + 2 \beta L _ { D V } L _ { p } \right) \left| s - s ^ { \dagger } \right| .
+$$
+
+Substitution $\begin{array} { r } { L _ { D V } = \frac { 2 L _ { r } + \frac { 4 \beta R _ { \mathrm { m a x } } L _ { p } } { 1 - \beta } } { 1 - 2 \beta L _ { p } } } \end{array}$ yields the statement of the Lemma.
+
+Proof of Theorem 4.2. From the Bellman equation (9), it is sufficient to show
+
+$$
+\frac { 1 } { \delta ( s ) ( 1 - \delta ( s ) ) } \geq 2 \beta ( \mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 1 ] - \mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 0 ] )
+$$
+
+We bound both sides separately. Since the policy satisfies $\delta ( s ) \leq { \bar { \delta } } ( s )$ for all states $s$ , and $\bar { \delta } ( s ) =$ 1exp(r(s))+1 ≤ 12 , the left hand side can be bounded from below as
+
+$$
+\frac { 1 } { \delta ( s ) ( 1 - \delta ( s ) ) } \geq \frac { 1 } { \bar { \delta } ( s ) ( 1 - \bar { \delta } ( s ) ) } \geq \frac { \left( \exp ( R _ { \operatorname* { m i n } } ) + 1 \right) ^ { 2 } } { \exp ( R _ { \operatorname* { m i n } } ) }
+$$
+
+Meanwhile, the righthand side can be bounded from above by Lemma B.1. Let $\begin{array} { r l } { L _ { D V } } & { { } = } \end{array}$ $\frac { 2 L _ { r } + \frac { 4 \beta R _ { \mathrm { m a x } } L _ { p } } { 1 - \beta } } { 1 - 2 \beta L _ { p } }$
+
+$$
+\begin{array} { r l } & { \quad 2 \beta \left| \mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 1 ] - \mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 0 ] \right| } \\ & { = 2 \beta \left| \int _ { s ^ { \prime } \in S } \mathrm { D } V _ { \delta } ( s ^ { \prime } ) ( p ( s ^ { \prime } | s , 1 ) - p ( s ^ { \prime } | s , 0 ) ) d s ^ { \prime } \right| } \\ & { = 2 \beta L _ { D V } \left| \int _ { s ^ { \prime } \in S } \frac { \mathrm { D } V _ { \delta } ( s ^ { \prime } ) } { L _ { D V } } ( p ( s ^ { \prime } | s , 1 ) - p ( s ^ { \prime } | s , 0 ) ) d s ^ { \prime } \right| } \\ & { \leq 2 \beta L _ { D V } \underset { \| f \| _ { L } \leq 1 } { \operatorname* { s u p } } \left\{ \left| \int _ { s ^ { \prime } \in S } f ( s ^ { \prime } ) ( p ( s ^ { \prime } | s , 1 ) - p ( s ^ { \prime } | s , 0 ) ) d s ^ { \prime } \right| \right\} } \\ & { = 2 \beta L _ { D V } K ( p ( \cdot | s , 1 ) , p ( \cdot | s , 0 ) ) \leq \frac { 2 \beta L _ { D } } { 1 - 2 \beta L _ { D } } \left( 2 L _ { r } + \frac { 4 \beta L _ { m a x } L _ { p } } { 1 - \beta } \right) } \end{array}
+$$
+
+From the condition of global convergence
+
+$$
+\frac { 2 \beta L _ { p } } { 1 - 2 \beta L _ { p } } \left( 2 L _ { r } + \frac { 4 \beta R _ { \mathrm { m a x } } L _ { p } } { 1 - \beta } \right) \leq \frac { \left( \exp ( R _ { \mathrm { m i n } } ) + 1 \right) ^ { 2 } } { \exp ( R _ { \mathrm { m i n } } ) }
+$$
+
+it follows that the inequality (10) is satisfied and the Bellman equation (9) implies that $\mathrm { D } ^ { 2 } V _ { \delta } ( s ) \le 0$ for all states $s \in S$ . □
+
+# B.3 OMITTED PROOF OF THEOREM 4.3
+
+For notation simplicity, we introduce notations $m$ and $M$ such that
+
+$$
+\begin{array} { l } { m = \displaystyle \frac { 1 } { 1 - \beta } \left( \frac { \left( \exp \left( R _ { \mathrm { m i n } } \right) + 1 \right) ^ { 2 } } { \exp ( R _ { \mathrm { m i n } } ) } - \frac { 2 \beta L _ { p } } { 1 - 2 \beta L _ { p } } \left( 2 L _ { r } + \frac { 4 \beta R _ { \mathrm { m a x } } L _ { p } } { 1 - \beta } \right) \right) } \\ { M = \displaystyle \frac { 1 } { ( 1 - \beta ) ^ { 2 } } \left( ( 1 - \beta ) \frac { \left( \exp \left( \frac { 1 } { 1 - \beta } R _ { \mathrm { m a x } } - \frac { \beta } { 2 } R _ { \mathrm { m i n } } \right) + 1 \right) ^ { 2 } } { \exp \left( \frac { 1 } { 1 - \beta } R _ { \mathrm { m a x } } - \frac { \beta } { 2 } R _ { \mathrm { m i n } } \right) } \right. } \\ { \displaystyle \left. + 2 \beta \left( \exp \left( \frac { 1 } { 1 - \beta } R _ { \mathrm { m a x } } - \frac { \beta } { 2 } R _ { \mathrm { m i n } } \right) + \frac { 2 \beta } { 1 - \beta } R _ { \mathrm { m a x } } - ( 1 + \beta ) R _ { \mathrm { m i n } } \right) \right) } \end{array}
+$$
+
+Theorem 4.3. Given a $( L _ { r } , L _ { p } )$ -Lipschitz MDP, which satisfies the condition for global convergence (6) and constants $m$ and $M$ defined above, for any step size $\begin{array} { r } { \alpha \leq \frac { 1 } { M } } \end{array}$ , the policy gradient initialized at the myopic policy $\bar { \delta }$ and updating as $\delta \gets \alpha \nabla _ { \delta } V _ { \delta }$ in the bounded Lipschitz policy space $\hat { \Delta }$ after $k$ iterations, it produces policy $\delta ^ { ( k ) }$ satisfying
+
+$$
+V _ { \tilde { \delta } } ( s ) - V _ { \delta ^ { ( k ) } } ( s ) \leq \frac { ( 1 - \alpha m ) ^ { k } } { \left( \exp ( R _ { \operatorname* { m i n } } ) + 1 \right) ^ { 2 } }
+$$
+
+at all $s \in S$
+
+Our analysis follows the standard steps establishing convergence of the conventional gradient descent algorithm which bounds the second Fréchet derivative of the value function $V _ { \delta }$ with respect to the policy $\delta$ from above and from below by $m$ and $M$ respectively.
+
+Lemma B.3. Given a $( L _ { r } , L _ { p } )$ -Lipschitz MDP, which satisfies the condition for global convergence (6), for all policies $\delta$ in the bounded Lipschitz policy space $\hat { \Delta }$ , for all states $s \in S$ , the second Fréchet derivative of the value function $V _ { \delta }$ with respect to the policy $\delta$ is upperbounded as
+
+$$
+\mathrm { D } ^ { 2 } V _ { \delta } ( s ) \leq - m .
+$$
+
+Proof. The Bellman equation (9) implies that
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { m a x } _ { s } \mathrm { D } ^ { 2 } V _ { \delta } ( s ) \leq \frac { 1 } { 1 - \beta } \left( - \operatorname* { m i n } _ { s } \frac { 1 } { \delta ( s ) ( 1 - \delta ( s ) ) } \right. } \\ { \displaystyle \qquad + \left. 2 \beta \operatorname* { m a x } _ { s } ( \mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 0 ] - \mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 1 ] ) \right) } \end{array}
+$$
+
+By the same argument as in Theorem 4.2,
+
+$$
+\begin{array} { r l } & { \underset { s } { \operatorname* { m i n } } \frac { 1 } { \delta ( s ) ( 1 - \delta ( s ) ) } \geq \frac { \left( \exp ( R _ { \operatorname* { m i n } } ) + 1 \right) ^ { 2 } } { \exp ( R _ { \operatorname* { m i n } } ) } } \\ & { \underset { s } { \operatorname* { m a x } } ( \mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s , 1 ] - \mathbf { E } _ { s ^ { \prime } } [ \mathrm { D } V _ { \delta } ( s ^ { \prime } ) | s ^ { \dagger } , 0 ] ) \leq \frac { 2 \beta L _ { p } } { 1 - 2 \beta L _ { p } } \left( 2 L _ { r } + \frac { 4 \beta R _ { \operatorname* { m a x } } L _ { p } } { 1 - \beta } \right) } \end{array}
+$$
+
+Thus, for all state $s \in S$ ,
+
+$$
+\mathrm { D } ^ { 2 } V _ { \delta } ( s ) \leq - m . \quad \bigsqcup
+$$
+
+Lemma B.4. Given a $( L _ { r } , L _ { p } )$ -Lipschitz MDP, which satisfies the condition for global convergence, for all policy $\delta$ in the bounded Lipschitz policy space $\hat { \Delta } ,$ , for all state $s \in S$ , the second derivative of the value function $V _ { \delta }$ with respect to the policy $\delta$ is is lowerbounded as
+
+$$
+\mathrm { D } ^ { 2 } V _ { \delta } ( s ) \geq - M .
+$$
+
+Proof. The Bellman equation (9) implies that
+
+$$
+\operatorname* { m i n } _ { s } \mathrm { D } ^ { 2 } V _ { \delta } ( s ) \geq \frac { 1 } { 1 - \beta } \left( - \operatorname* { m a x } _ { s } \frac { 1 } { \delta ( s ) ( 1 - \delta ( s ) ) } \right.
+$$
+
+By restricting policy to the bounded Lipschitz policy space $\hat { \Delta }$ we bound
+
+$$
+\operatorname* { m a x } _ { s } \frac { 1 } { \delta ( s ) ( 1 - \delta ( s ) ) } \leq \operatorname* { m a x } _ { s } \frac { 1 } { \underline { { \delta } } ( s ) ( 1 - \underline { { \delta } } ( s ) ) } \leq \frac { \left( \exp \left( \frac { 1 } { 1 - \beta } R _ { \operatorname* { m a x } } - \frac { \beta } { 2 } R _ { \operatorname* { m i n } } \right) + 1 \right) ^ { 2 } } { \exp ( \frac { 1 } { 1 - \beta } R _ { \operatorname* { m a x } } - \frac { \beta } { 2 } R _ { \operatorname* { m i n } } ) }
+$$
+
+Provided
+
+$$
+\begin{array} { r l } & { \underset { s } { \operatorname* { m i n } } V _ { \delta } ( s ) \geq \frac { R _ { \operatorname* { m i n } } } { 2 } } \\ & { \underset { s } { \operatorname* { m a x } } V _ { \delta } ( s ) \leq \frac { R _ { \operatorname* { m a x } } } { 1 - \beta } } \\ & { \underset { s } { \operatorname* { m i n } } \left( \log \left( \frac { 1 - \delta ( s ) } { \delta ( s ) } \right) - r ( s ) \right) \geq \underset { s } { \operatorname* { m i n } } \left( \log \left( \frac { 1 - \bar { \delta } ( s ) } { \delta ( s ) } \right) - r ( s ) \right) = 0 } \\ & { \underset { s } { \operatorname* { m a x } } \left( \log \left( \frac { 1 - \delta ( s ) } { \delta ( s ) } \right) - r ( s ) \right) \leq \underset { s } { \operatorname* { m a x } } \log \left( \frac { 1 - \bar { \delta } ( s ) } { \bar { \delta } ( s ) } \right) - \underset { s } { \operatorname* { m i n } } r ( s ) } \\ & { \qquad \quad \leq \exp \left( \frac { 1 } { 1 - \beta } R _ { \operatorname* { m a x } } - \frac { \beta } { 2 } R _ { \operatorname* { m i n } } \right) - R _ { \operatorname* { m i n } } } \end{array}
+$$
+
+it follows from Bellman equation (8) that
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { m i n } _ { s } \mathrm { D } V _ { \delta } ( s ) \geq \frac { 1 } { 1 - \beta } \left( \operatorname* { m i n } _ { s } \left( \log \left( \frac { 1 - \delta ( s ) } { \delta ( s ) } \right) - r ( s ) \right) + \beta ( \operatorname* { m i n } _ { s } V _ { \delta } ( s ) - \operatorname* { m a x } _ { s } V _ { \delta } ( s ) ) \right) } \\ { \displaystyle \qquad \geq \frac { \beta } { 1 - \beta } \left( \frac { R _ { \operatorname* { m i n } } } { 2 } - \frac { R _ { \operatorname* { m a x } } } { 1 - \beta } \right) } \\ { \displaystyle \operatorname* { m a x } _ { s } \mathrm { D } V _ { \delta } ( s ) \leq \frac { 1 } { 1 - \beta } \left( \operatorname* { m a x } _ { s } \left( \log \left( \frac { 1 - \delta ( s ) } { \delta ( s ) } \right) - r ( s ) \right) + \beta ( \operatorname* { m a x } _ { s } V _ { \delta } ( s ) - \operatorname* { m i n } _ { s } V _ { \delta } ( s ) ) \right) } \\ { \displaystyle \qquad \leq \frac { 1 } { 1 - \beta } \left( \exp \left( \frac { 1 } { 1 - \beta } R _ { \operatorname* { m a x } } - \frac { \beta } { 2 } R _ { \operatorname* { m i n } } \right) + \frac { \beta } { 1 - \beta } R _ { \operatorname* { m a x } } - \frac { 2 + \beta } { 2 } R _ { \operatorname* { m i n } } \right) } \end{array}
+$$
+
+Thus, for all state $s \in S$ ,
+
+$$
+\mathrm { D } ^ { 2 } V _ { \delta } ( s ) \geq - M . \quad \bigsqcup
+$$
+
+Proof of Theorem 4.3. The convergence rate guarantee follows from Lemma B.3 and Lemma B.4, under the standard arguments for the gradient descent algorithm for $m$ -strongly concave and $M$ - smooth (i.e., $M$ -Lipschitz gradient) functions (cf. Bansal and Gupta, 2017). □
+
+# C MORE RESULTS IN SECTION 5
+
+<table><tr><td>Algorithm 1&quot;Lazy projection”, (π1,...,πN): thresholds corresponding to discretized state space (S1, ..,SN); L(.): logistic function; policy δj = L(πj); α: step size; T: termination tolerance</td></tr><tr><td>（0） ←u(s1),.,π 1: （0） ← u(s n) // Initialize π(O) at the myopic policy T while maxi [DV(e）(si)≤Tdo 2:</td></tr><tr><td>(k*) (k-1) 3: 下 ↑</td></tr><tr><td>(k) (k)) the closest monotone thresholds of(π( (k*) (k*)) 4: (i 个 ) // Lazy projection T N πN ， (k) (k）) 5: return πN</td></tr></table>
+
+We list the convergence of gradient descent and its derivative, second derivative at smaller discount factor $\beta = 0 . 9$ .
+
+![](images/818ad2e166dc674c2f64fd20e2cf93f602748d9aa51c8ca367cf330849e1356f.jpg)  
+Figure 3: Convergence of gradient descent, discount factor $\beta = 0 . 9$
+
+![](images/b2ed97dbdfb18a692db17c075415d3dc84eefad284a4a76be02da629e5284013.jpg)  
+Figure 4: Performance of the norm $\mathrm { m a x } _ { i }$ $\left| \mathrm Ḋ V Ḍ _ { \delta } ( s _ { i } ) \right|$ and the second derivative maxi $ { \left| \mathrm { D } ^ { 2 } V _ { \delta } ( s _ { i } ) \right| }$ , discount factor $\beta = 0 . 9$

parse/train/H1meywxRW/H1meywxRW.md

@@ -0,0 +1,239 @@
+# DCN+: MIXED OBJECTIVE AND DEEP RESIDUAL COATTENTION FOR QUESTION ANSWERING
+
+Caiming Xiong∗, Victor Zhong∗, Richard Socher
+
+Salesforce Research   
+Palo Alto, CA 94301, USA   
+{cxiong, vzhong, rsocher}@salesforce.com
+
+# ABSTRACT
+
+Traditional models for question answering optimize using cross entropy loss, which encourages exact answers at the cost of penalizing nearby or overlapping answers that are sometimes equally accurate. We propose a mixed objective that combines cross entropy loss with self-critical policy learning. The objective uses rewards derived from word overlap to solve the misalignment between evaluation metric and optimization objective. In addition to the mixed objective, we improve dynamic coattention networks (DCN) with a deep residual coattention encoder that is inspired by recent work in deep self-attention and residual networks. Our proposals improve model performance across question types and input lengths, especially for long questions that requires the ability to capture long-term dependencies. On the Stanford Question Answering Dataset, our model achieves state-of-the-art results with $7 5 . 1 \%$ exact match accuracy and $8 3 . 1 \%$ F1, while the ensemble obtains $7 8 . 9 \%$ exact match accuracy and $8 6 . 0 \%$ F1.
+
+# 1 INTRODUCTION
+
+Existing state-of-the-art question answering models are trained to produce exact answer spans for a question and a document. In this setting, a ground truth answer used to supervise the model is defined as a start and an end position within the document. Existing training approaches optimize using cross entropy loss over the two positions. However, this suffers from a fundamental disconnect between the optimization, which is tied to the position of a particular ground truth answer span, and the evaluation, which is based on the textual content of the answer. This disconnect is especially harmful in cases where answers that are textually similar to, but distinct in positions from, the ground truth are penalized in the same fashion as answers that are textually dissimilar. For example, suppose we are given the sentence “Some believe that the Golden State Warriors team of 2017 is one of the greatest teams in NBA history”, the question “which team is considered to be one of the greatest teams in NBA history”, and a ground truth answer of “the Golden State Warriors team of 2017”. The span “Warriors” is also a correct answer, but from the perspective of traditional cross entropy based training it is no better than the span “history”.
+
+To address this problem, we propose a mixed objective that combines traditional cross entropy loss over positions with a measure of word overlap trained with reinforcement learning. We obtain the latter objective using self-critical policy learning in which the reward is based on word overlap between the proposed answer and the ground truth answer. Our mixed objective brings two benefits: (i) the reinforcement learning objective encourages answers that are textually similar to the ground truth answer and discourages those that are not; (ii) the cross entropy objective significantly facilitates policy learning by encouraging trajectories that are known to be correct. The resulting objective is one that is both faithful to the evaluation metric and converges quickly in practice.
+
+In addition to our mixed training objective, we extend the Dynamic Coattention Network (DCN) by Xiong et al. (2017) with a deep residual coattention encoder. This allows the network to build richer representations of the input by enabling each input sequence to attend to previous attention contexts. Vaswani et al. (2017) show that the stacking of attention layers helps model long-range dependencies. We merge coattention outputs from each layer by means of residual connections to reduce the length of signal paths. He et al. (2016) show that skip layer connections facilitate signal propagation and alleviate gradient degradation.
+
+![](images/699cf74e131ca40810b83acce546596d257f708b1b3d1ec693618e6b03cea3d7.jpg)  
+Figure 1: Deep residual coattention encoder.
+
+The combination of the deep residual coattention encoder and the mixed objective leads to higher performance across question types, question lengths, and answer lengths on the Stanford Question Answering Dataset (SQuAD) (Rajpurkar et al., 2016) compared to our DCN baseline. The improvement is especially apparent on long questions, which require the model to capture long-range dependencies between the document and the question. Our model, which we call $\mathrm { D C N + }$ , achieves state-of-the-art results on SQuAD, with $7 5 . 1 \%$ exact match accuracy and $8 3 . 1 \%$ F1. When ensembled, the $\mathrm { D C N + }$ obtains $78 . 9 \%$ exact match accuracy and $8 6 . 0 \%$ F1.
+
+# 2 $\mathrm { D C N + }$
+
+We consider the question answering task in which we are given a document and a question, and are asked to find the answer in the document. Our model is based on the DCN by Xiong et al. (2017), which consists of a coattention encoder and a dynamic decoder. The encoder first encodes the question and the document separately, then builds a codependent representation through coattention. The decoder then produces a start and end point estimate given the coattention. The DCN decoder is dynamic in the sense that it iteratively estimates the start and end positions, stopping when estimates between iterations converge to the same positions or when a predefined maximum number of iterations is reached. We make two significant changes to the DCN by introducing a deep residual coattention encoder and a mixed training objective that combines cross entropy loss from maximum likelihood estimation and reinforcement learning rewards from self-critical policy learning.
+
+# 2.1 DEEP RESIDUAL COATTENTION ENCODER
+
+Because it only has a single-layer coattention encoder, the DCN is limited in its ability to compose complex input representations. Vaswani et al. (2017) proposed stacked self-attention modules to facilitate signal traversal. They also showed that the network’s ability to model long-range dependencies can be improved by reducing the length of signal paths. We propose two modifications to the coattention encoder to leverage these findings. First, we extend the coattention encoder with self-attention by stacking coattention layers. This allows the network to build richer representations over the input. Second, we merge coattention outputs from each layer with residual connections. This reduces the length of signal paths. Our encoder is shown in Figure 1.
+
+Suppose we are given a document of $m$ words and a question of $n$ words. Let $L ^ { D } \in \mathbb { R } ^ { e \times m }$ and $L ^ { \mathcal { Q } ^ { \star } } \in \mathbb { R } ^ { e \times n }$ respectively denote the word embeddings for the document and the question, where $e$ is the dimension of the word embeddings. We obtain document encodings $E _ { 1 } ^ { \hat { D } }$ and question encodings $E _ { 1 } ^ { Q }$ through a bidirectional Long Short-Term Memory Network (LSTM) (Hochreiter & Schmidhuber, 1997), where we use integer subscripts to denote the coattention layer number.
+
+$$
+\begin{array} { r c l } { E _ { 1 } ^ { D } } & { = } & { \mathrm { b i L S T M } _ { 1 } \left( L ^ { D } \right) \in \mathbb { R } ^ { h \times ( m + 1 ) } } \\ { E _ { 1 } ^ { Q } } & { = } & { \operatorname { t a n h } \big ( W \mathrm { \ b i L S T M } _ { 1 } \left( L ^ { Q } \right) + b \big ) \in \mathbb { R } ^ { h \times ( n + 1 ) } } \end{array}
+$$
+
+Here, $h$ denotes the hidden state size and the $+ 1$ indicates the presence of an additional sentinel word which allows the coattention to not focus on any part of the input. Like the original DCN, we add a non-linear transform to the question encoding.
+
+We compute the affinity matrix between the document and the question as $A \ = \ \left( E _ { 1 } ^ { D } \right) ^ { \boldsymbol { \mathsf { T } } } E _ { 1 } ^ { Q } \ \in$ $\mathbb { R } ^ { ( m + 1 ) \times ( n + 1 ) }$ . Let softmax $( X )$ denote the softmax operation over the matrix $X$ that normalizes $X$ column-wise. The document summary vectors and question summary vectors are computed as
+
+$$
+\begin{array} { r c l } { S _ { 1 } ^ { D } } & { = } & { E _ { 1 } ^ { Q } \operatorname { s o f t m a x } \left( A ^ { \boldsymbol { \mathsf { T } } } \right) \in \mathbb { R } ^ { h \times ( m + 1 ) } } \\ { S _ { 1 } ^ { Q } } & { = } & { E _ { 1 } ^ { D } \operatorname { s o f t m a x } \left( A \right) \in \mathbb { R } ^ { h \times ( n + 1 ) } } \end{array}
+$$
+
+We define the document coattention context as follows. Note that we drop the dimension corresponding to the sentinel vector – it has already been used during the summary computation and is not a potential position candidate for the decoder.
+
+$$
+\begin{array} { r c l } { C _ { 1 } ^ { D } } & { = } & { S _ { 1 } ^ { Q } \operatorname { s o f t m a x } \left( A ^ { \intercal } \right) \in \mathbb { R } ^ { h \times m } } \end{array}
+$$
+
+We further encode the summaries using another bidirectional LSTM.
+
+$$
+\begin{array} { r l r } { E _ { 2 } ^ { D } } & { = } & { \mathrm { b i L S T M _ { 2 } } \left( S _ { 1 } ^ { D } \right) \in \mathbb { R } ^ { 2 h \times m } } \\ { E _ { 2 } ^ { Q } } & { = } & { \mathrm { b i L S T M _ { 2 } } \left( S _ { 1 } ^ { Q } \right) \in \mathbb { R } ^ { 2 h \times n } } \end{array}
+$$
+
+Equation 3 to equation 5 describe a single coattention layer. We compute the second coattention layer in a similar fashion. Namely, let coattn denote a multi-valued mapping whose inputs are the two input sequences $E _ { 1 } ^ { D }$ and $E _ { 1 } ^ { Q }$ . We have
+
+$$
+{ \begin{array} { l l l } { \operatorname { c o a t t n _ { 1 } } \left( E _ { 1 } ^ { D } , E _ { 1 } ^ { Q } \right) } & { \to } & { S _ { 1 } ^ { D } , S _ { 1 } ^ { Q } , C _ { 1 } ^ { D } } \\ & & { \operatorname { c o a t t n _ { 2 } } \left( E _ { 2 } ^ { D } , E _ { 2 } ^ { Q } \right) } \\ { \operatorname { c o a t t n _ { 2 } } \left( E _ { 2 } ^ { D } , E _ { 2 } ^ { Q } \right) } & { \to } & { S _ { 2 } ^ { D } , S _ { 2 } ^ { Q } , C _ { 2 } ^ { D } } \end{array} }
+$$
+
+The output of our encoder is then obtained as
+
+$$
+U = \mathrm { b i L S T M } \left( \mathrm { c o n c a t } \left( E _ { 1 } ^ { D } ; E _ { 2 } ^ { D } ; S _ { 1 } ^ { D } ; S _ { 2 } ^ { D } ; C _ { 1 } ^ { D } ; C _ { 2 } ^ { D } \right) \right) \in \mathbb { R } ^ { 2 h \times m }
+$$
+
+where concat $( A , B )$ denotes the concatenation between the matrices $A$ and $B$ along the first dimension.
+
+This encoder is different than the original DCN in its depth and its use of residual connections. We use not only the output of the deep coattention network $\bar { C } _ { 2 } ^ { D }$ as input to the final bidirectional LSTM, but add skip connections to initial encodings $E _ { 1 } ^ { D }$ , $E _ { 2 } ^ { D }$ 2, summary vectors $S _ { 1 } ^ { D } , S _ { 2 } ^ { D }$ , and coattention context $C _ { 1 } ^ { \bar { D } }$ . This is akin to transformer networks (Vaswani et al., 2017), which achieved stateof-the-art results on machine translation using deep self-attention layers to help model long-range dependencies, and residual networks (He et al., 2016), which achieved state-of-the-art results in image classification through the addition of skip layer connections to facilitate signal propagation and alleviate gradient degradation.
+
+The DCN produces a distribution over the start position of the answer and a distribution over the end position of the answer. Let $s$ and $e$ denote the respective start and end points of the ground truth answer. Because the decoder of the DCN is dynamic, we denote the start and end distributions produced at the tth decoding step by $p _ { t } ^ { \mathrm { s t a r t } } \in \mathbb { R } ^ { m }$ and $p _ { t } ^ { \mathrm { e n d } } \in \mathbb { R } ^ { m }$ . For convenience, we denote the greedy estimate of the start and end positions by the model at the tth decoding step by $s _ { t }$ and $e _ { t }$ . Moreover, let $\Theta$ denote the parameters of the model.
+
+Similar to other question answering models, the DCN is supervised using the cross entropy loss on the start position distribution and the end position distribution:
+
+$$
+l _ { c e } ( \Theta ) = - \sum _ { t } \left( \log p _ { t } ^ { \mathrm { s t a r t } } \left( s \mid s _ { t - 1 } , e _ { t - 1 } ; \Theta \right) + \log p _ { t } ^ { \mathrm { e n d } } \left( e \mid s _ { t - 1 } , e _ { t - 1 } ; \Theta \right) \right)
+$$
+
+Equation 11 states that the model accumulates a cross entropy loss over each position during each decoding step given previous estimates of the start and end positions.
+
+The question answering task consists of two evaluation metrics. The first, exact match, is a binary score that denotes whether the answer span produced by the model has exact string match with the ground truth answer span. The second, F1, computes the degree of word overlap between the answer span produced by the model and the ground truth answer span. We note that there is a disconnect between the cross entropy optimization objective and the evaluation metrics. For example, suppose we are given the answer estimates $A$ and $B$ , neither of which match the ground truth positions. However, $A$ has an exact string match with the ground truth answer whereas $B$ does not. The cross entropy objective penalizes $A$ and $B$ equally, despite the former being correct under both evaluation metrics. In the less extreme case where $A$ does not have exact match but has some degree of word overlap with the ground truth, the F1 metric still prefers $A$ over $B$ despite its wrongly predicted positions.
+
+We encode this preference using reinforcement learning, using the F1 score as the reward function. Let $\hat { s _ { t } } \sim p _ { t } ^ { \mathrm { s t a r t } }$ and $\hat { e _ { t } } \sim p _ { t } ^ { \mathrm { s t a r t } }$ denote the sampled start and end positions from the estimated distributions at decoding step $t$ . We define a trajectory $\hat { \tau }$ as a sequence of sampled start and end points $\hat { s _ { t } }$ and $\hat { e _ { t } }$ through all $T$ decoder time steps. The reinforcement learning objective is then the negative expected rewards $R$ over trajectories.
+
+$$
+\begin{array} { r l r } { l _ { r l } \left( \Theta \right) } & { = } & { - \mathbb { E } _ { \hat { r } \sim p _ { \tau } } \left[ R \left( s , e , \hat { s } _ { T } , \hat { e } _ { T } ; \Theta \right) \right] } \\ & { \approx } & { - \mathbb { E } _ { \hat { \tau } \sim p _ { \tau } } \left[ F _ { 1 } \left( \mathrm { a n s } \left( \hat { \mathrm { s } } _ { \mathrm { T } } , \hat { \mathrm { e } } _ { \mathrm { T } } \right) , \mathrm { a n s } \left( \mathrm { s } , \mathrm { e } \right) \right) - F _ { 1 } \left( \mathrm { a n s } \left( \mathrm { s } _ { \mathrm { T } } , \mathrm { e } _ { \mathrm { T } } \right) , \mathrm { a n s } \left( \mathrm { s } , \mathrm { e } \right) \right) \right] } \end{array}
+$$
+
+We use $F _ { 1 }$ to denote the F1 scoring function and ans $( \mathrm { s } , \mathrm { e } )$ to denote the answer span retrieved using the start point $s$ and end point $e$ . In equation 13, instead of using only the F1 word overlap as the reward, we subtract from it a baseline. Greensmith et al. (2001) show that a good baseline reduces the variance of gradient estimates and facilitates convergence. In our case, we employ a self-critic (Konda $\&$ Tsitsiklis, 1999) that uses the F1 score produced by the current model during greedy inference without teacher forcing.
+
+For ease of notation, we abbreviate $R \left( s , e , \hat { s } _ { T } , \hat { e } _ { T } ; \Theta \right)$ as $R$ . As per Sutton et al. (1999) and Schulman et al. (2015), the expected gradient of a non-differentiable reward function can be computed as
+
+$$
+\begin{array} { r c l } { \displaystyle \mathcal { T } _ { \Theta } l _ { r l } ( \Theta ) } & { = } & { \displaystyle - \nabla _ { \Theta } ( \mathbb { E } _ { \hat { r } \sim p _ { r } } [ R ] ) } \\ & { = } & { \displaystyle - \mathbb { E } _ { \hat { r } \sim p _ { r } } [ R \nabla _ { \Theta } \log p _ { r } ( \boldsymbol { r } ; \Theta ) ] } \\ & { = } & { \displaystyle - \mathbb { E } _ { \hat { r } \sim p _ { r } } [ R \nabla _ { \Theta } ( \sum _ { t } ^ { T } ( \log p _ { t } ^ { \mathrm { s t a r t } } ( \hat { s } _ { t } | \hat { s } _ { t - 1 } , \hat { e } _ { t - 1 } ; \Theta ) + \log p _ { t } ^ { \mathrm { e n d } } ( \hat { e } _ { t } | \hat { s } _ { t - 1 } , \hat { e } _ { t - 1 } ; \Theta ) ) ) ] } \\ & { \approx } & { \displaystyle - R \nabla _ { \Theta } ( \sum _ { t } ^ { T } ( \log p _ { t } ^ { \mathrm { s t a r t } } ( \hat { s } _ { t } | \hat { s } _ { t - 1 } , \hat { e } _ { t - 1 } ; \Theta ) + \log p _ { t } ^ { \mathrm { e n d } } ( \hat { e } _ { t } | \hat { s } _ { t - 1 } , \hat { e } _ { t - 1 } ; \Theta ) ) ) \qquad ( \mathrm { l } \Theta ) \mathrm { ~ } } \end{array}
+$$
+
+![](images/6004dd53bb1fbb962861f9aee8a80aa6706f83f2e92cdbcdfe5fd35c7b41eefa.jpg)  
+Figure 2: Computation of the mixed objective.
+
+In equation 16, we approximate the expected gradient using a single Monte-Carlo sample $\tau$ drawn from $p _ { \tau }$ . This sample trajectory $\tau$ contains the start and end positions $\hat { s } _ { t }$ and $\hat { e } _ { t }$ sampled during all decoding steps.
+
+One of the key problems in applying RL to natural language processing is the discontinuous and discrete space the agent must explore in order to find a good policy. For problems with large exploration space, RL approaches tend to be applied as fine-tuning steps after a maximum likelihood model has already been trained (Paulus et al., 2017; Wu et al., 2016). The resulting model is constrained in its exploration during fine-tuning because it is biased by heavy pretraining. We instead treat the optimization problem as a multi-task learning problem. The first task is to optimize for positional match with the ground truth answer using the the cross entropy objective. The second task is to optimize for word overlap with the ground truth answer with the self-critical reinforcement learning objective. In a similar fashion to Kendall et al. (2017), we combine the two losses using homoscedastic uncertainty as task-dependent weightings.
+
+$$
+l = \frac { 1 } { 2 \sigma _ { c e } ^ { 2 } } l _ { c e } \left( \Theta \right) + \frac { 1 } { 2 \sigma _ { r l } ^ { 2 } } l _ { r l } \left( \Theta \right) + \log \sigma _ { c e } ^ { 2 } + \log \sigma _ { r l } ^ { 2 }
+$$
+
+Here, $\sigma _ { c e }$ and $\sigma _ { r l }$ are learned parameters. The gradient of the cross entropy objective can be derived using straight-forward backpropagation. The gradient of the self-critical reinforcement learning objective is shown in equation 16. Figure 2 illustrates how the mixed objective is computed. In practice, we find that adding the cross entropy task significantly facilitates policy learning by pruning the space of candidate trajectories - without the former, it is very difficult for policy learning to converge due to the large space of potential answers, documents, and questions.
+
+# 3 EXPERIMENTS
+
+We train and evaluate our model on the Stanford Question Answering Dataset (SQuAD). We show our test performance of our model against other published models, and demonstrate the importance of our proposals via ablation studies on the development set. To preprocess the corpus, we use the reversible tokenizer from Stanford CoreNLP (Manning et al., 2014). For word embeddings, we use GloVe embeddings pretrained on the 840B Common Crawl corpus (Pennington et al., 2014) as well as character ngram embeddings by Hashimoto et al. (2017). In addition, we concatenate these embeddings with context vectors (CoVe) trained on WMT (McCann et al., 2017). For out of vocabulary words, we set the embeddings and context vectors to zero. We perform word dropout on the document which zeros a word embedding with probability 0.075. In addition, we swap the first maxout layer of the highway maxout network in the DCN decoder with a sparse mixture of experts layer (Shazeer et al., 2017). This layer is similar to the maxout layer, except instead of taking the top scoring expert, we take the top $k = 2$ expert. The model is trained using ADAM (Kingma & Ba,
+
+2014) with default hyperparameters. Hyperparameters of our model are identical to the DCN. We implement our model using PyTorch.
+
+3.1 RESULTS   
+
+<table><tr><td></td><td colspan="2">Single Model Dev</td><td colspan="2">Single Model Test</td><td colspan="2">Ensemble Test</td></tr><tr><td>Model</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td></tr><tr><td>DCN+ (ours)</td><td>74.5%</td><td>83.1%</td><td>75.1%</td><td>83.1%</td><td>78.9%</td><td>86.0%</td></tr><tr><td>rnet</td><td>72.3%</td><td>80.6%</td><td>72.3%</td><td>80.7%</td><td>76.9%</td><td>84.0%</td></tr><tr><td>DCN w/ CoVe (baseline)</td><td>71.3%</td><td>79.9%</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Mnemonic Reader</td><td>70.1%</td><td>79.6%</td><td>69.9%</td><td>79.2%</td><td>73.7%</td><td>81.7%</td></tr><tr><td>Document Reader</td><td>69.5%</td><td>78.8%</td><td>70.0%</td><td>79.0%</td><td>1</td><td>1</td></tr><tr><td>FastQA</td><td>70.3%</td><td>78.5%</td><td>70.8%</td><td>78.9%</td><td>1</td><td>一</td></tr><tr><td>ReasoNet</td><td>1</td><td>1</td><td>69.1%</td><td>78.9%</td><td>73.4%</td><td>81.8%</td></tr><tr><td>SEDT</td><td>67.9%</td><td>77.4%</td><td>68.5%</td><td>78.0%</td><td>73.0%</td><td>80.8%</td></tr><tr><td>BiDAF</td><td>67.7%</td><td>77.3%</td><td>68.0%</td><td>77.3%</td><td>73.7%</td><td>81.5%</td></tr><tr><td>DCN</td><td>65.4%</td><td>75.6%</td><td>66.2%</td><td>75.9%</td><td>71.6%</td><td>80.4%</td></tr></table>
+
+Table 1: Test performance on SQuAD. The papers are as follows: rnet (Microsoft Asia Natural Language Computing Group, 2017), SEDT (Liu et al., 2017), BiDAF (Seo et al., 2017), DCN w/ CoVe (McCann et al., 2017), ReasoNet (Shen et al., 2017), Document Reader (Chen et al., 2017), FastQA (Weissenborn et al., 2017), DCN (Xiong et al., 2017). The CoVe authors did not submit their model, which we use as our baseline, for SQuAD test evaluation.
+
+The performance of our model is shown in Table 1. Our model achieves state-of-the-art results on SQuAD dataset with $7 5 . 1 \%$ exact match accuracy and $8 3 . 1 \%$ F1. When ensembled, our model obtains $78 . 9 \%$ exact match accuracy and $8 6 . 0 \%$ F1. To illustrate the effectiveness of our proposals, we use the DCN with context vectors as a baseline (McCann et al., 2017). This model is identical to the DCN by Xiong et al. (2017), except that it augments the word representations with context vectors trained on WMT16.
+
+Comparison to baseline DCN with CoVe. $\mathrm { D C N + }$ outperforms the baseline by $3 . 2 \%$ exact match accuracy and $3 . 2 \%$ F1 on the SQuAD development set. Figure 3 shows the consistent performance gain of $\mathrm { D C N + }$ over the baseline across question types, question lengths, and answer lengths. In particular, $\mathrm { D C N + }$ provides a significant advantage for long questions.
+
+![](images/fbb0659f94d67783dc74da5e903666db4118a5bbafdc6f829825c1345f842277.jpg)  
+Figure 3: Performance comparison between $\mathrm { D C N + }$ and the baseline DCN with CoVe on the SQuAD development set.
+
+Ablation study. The contributions of each part of our model are shown in Table 2. We note that the deep residual coattention yielded the highest contribution to model performance, followed by the mixed objective. The sparse mixture of experts layer in the decoder added minor improvements to the model performance.
+
+<table><tr><td>Model</td><td>EM</td><td>△EM</td><td>F1</td><td>△F1</td></tr><tr><td>DCN+ (ours)</td><td>74.5%</td><td>1</td><td>83.1%</td><td>1</td></tr><tr><td>- Deep residual coattention</td><td>73.1%</td><td>-1.4%</td><td>81.5%</td><td>-1.6%</td></tr><tr><td> - Mixed objective</td><td>73.8%</td><td>-0.7%</td><td>82.1%</td><td>-1.0%</td></tr><tr><td>- Mixture of experts</td><td>74.0%</td><td>-0.5%</td><td>82.4%</td><td>-0.7%</td></tr><tr><td>DCN w/ CoVe (baseline)</td><td>71.3%</td><td>-3.2%</td><td>79.9%</td><td>-3.2%</td></tr></table>
+
+Table 2: Ablation study on the development set of SQuAD.
+
+![](images/d28d264cde55c7154152e799bb4c6411d9033cadfa8609e875ca04717a0428f6.jpg)  
+Figure 4: Training curve of $\mathrm { D C N + }$ with and without reinforcement learning. In the latter case, only the cross entropy objective is used. The mixed objective initially performs worse as it begins policy learning from scratch, but quickly outperforms the cross entropy model.
+
+Mixed objective convergence. The training curves for $\mathrm { D C N + }$ with reinforcement learning and $\mathrm { D C N + }$ without reinforcement learning are shown in Figure 4 to illustrate the effectiveness of our proposed mixed objective. In particular, we note that without mixing in the cross entropy loss, it is extremely difficult to learn the policy. When we combine the cross entropy loss with the reinforcement learning objective, we find that the model initially performs worse early on as it begins policy learning from scratch (shown in Figure 4b). However, with the addition of cross entropy loss, the model quickly learns a reasonable policy and subsequently outperforms the purely cross entropy model (shown in Figure 4a).
+
+Sample predictions. Figure 5 compares predictions by $\mathrm { D C N + }$ and by the baseline on the development set. Both models retrieve answers that have sensible entity types. For example, the second example asks for “what game” and both models retrieve an American football game; the third example asks for “type of Turing machine” and both models retrieve a type of turing machine. We find, however, that $\mathrm { D C N + }$ consistently make less mistakes on finding the correct entity. This is especially apparent in the examples we show, which contain several entities or candidate answers of the correct type. In the first example, Gasquet wrote about the plague and called it “Great Pestilence”. While he likely did think of the plague as a “great pestilence”, the phrase “suggested that it would appear to be some form of ordinary Eastern or bubonic plague” provides evidence for the correct answer – “some form of ordinary Eastern or bubonic plague”. Similarly, the second example states that Thomas Davis was injured in the “NFC Championship Game”, but the game he insisted on playing in is the “Super Bowl”. Finally, “multi-tape” and “single-tape” both appear in the sentence that provides provenance for the answer to the question. However, it is the “single-tape” Turing machine that implies quadratic time. In these examples, $\mathrm { D C N + }$ finds the correct entity out of ones that have the right type whereas the baseline does not.
+
+# 4 RELATED WORK
+
+Neural models for question answering. Current state-of-the-art approaches for question answering over unstructured text tend to be neural approaches. Wang & Jiang (2017) proposed one of the first conditional attention mechanisms in the Match-LSTM encoder. Coattention (Xiong et al., 2017), bidirectional attention flow (Seo et al., 2017), and self-matching attention (Microsoft Asia Natural Language Computing Group, 2017) all build codependent representations of the question and the document. These approaches of conditionally encoding two sequences are widely used in
+
+What did Gasquet think the plague was?
+
+The historian Francis Aidan Gasquet wrote about the 'Great Pestilence' in 1893 and suggested that "it would appear to be some form of the ordinary Eastern or bubonic plague". He was able to adopt the epidemiology of the bubonic plague for the Black Death for the second edition in 1908, implicating rats and fleas in the process, and his interpretation was widely accepted for other ancient and medieval epidemics, such as the Justinian plague that was prevalent in the Eastern Roman Empire from 541 to 700 CE.
+
+What game did Thomas Davis say he would play in, despite breaking a bone earlier on?
+
+Carolina suffered a major setback when Thomas Davis, an 11-year veteran who had already overcome three ACL tears in his career, went down with a broken arm in the NFC Championship Game. Despite this, he insisted he would still find a way to play in the Super Bowl. His prediction turned out to be accurate.
+
+A language solved in quadratic time implies the use of what type of Turing machine?
+
+But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language $\{ \times \times | \times$ is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.
+
+Figure 5: Predictions by $\mathrm { D C N + }$ (red) and DCN with CoVe (blue) on the SQuAD development set.
+
+question answering. After building codependent encodings, most models predict the answer by generating the start position and the end position corresponding to the estimated answer span. The generation process utilizes a pointer network (Vinyals et al., 2015) over the positions in the document. Xiong et al. (2017) also introduced the dynamic decoder, which iteratively proposes answers by alternating between start position and end position estimates, and in some cases is able to recover from initially erroneous predictions.
+
+Neural attention models. Neural attention models saw early adoption in machine translation (Bahdanau et al., 2015) and has since become to de-facto architecture for neural machine translation models. Self-attention, or intra-attention, has been applied to language modeling, sentiment analysis, natural language inference, and abstractive text summarization (Chen et al., 2017; Paulus et al., 2017). Vaswani et al. (2017) extended this idea to a deep self-attentional network which obtained state-of-the-art results in machine translation. Coattention, which builds codependent representations of multiple inputs, has been applied to visual question answering (Lu et al., 2016). Xiong et al. (2017) introduced coattention for question answering. Bidirectional attention flow (Seo et al., 2017) and self-matching attention (Microsoft Asia Natural Language Computing Group, 2017) also build codependent representations between the question and the document.
+
+Reinforcement learning in NLP. Many tasks in natural language processing have evaluation metrics that are not differentiable. Dethlefs & Cuayahuitl ´ (2011) proposed a hierarchical reinforcement learning technique for generating text in a simulated way-finding domain. Narasimhan et al. (2015) applied deep Q-networks to learn policies for text-based games using game rewards as feedback. Li et al. (2016) introduced a neural conversational model trained using policy gradient methods, whose reward function consisted of heuristics for ease of answering, information flow, and semantic coherence. Bahdanau et al. (2017) proposed a general actor-critic temporal-difference method for sequence prediction, performing metric optimization on language modeling and machine translation. Direct word overlap metric optimization has also been applied to summarization (Paulus et al., 2017), and machine translation (Wu et al., 2016).
+
+# 5 CONCLUSION
+
+We introduced $\mathrm { D C N + }$ , an state-of-the-art question answering model with deep residual coattention trained using a mixed objective that combines cross entropy supervision with self-critical policy learning. We showed that our proposals improve model performance across question types, question lengths, and answer lengths on the Stanford Question Answering Dataset ( SQuAD). On SQuAD, the $\mathrm { D C N + }$ achieves $7 5 . 1 \%$ exact match accuracy and $8 3 . 1 \%$ F1. When ensembled, the $\mathrm { D C N + }$ obtains $7 8 . 9 \%$ exact match accuracy and $8 6 . 0 \%$ F1.
+
+# REFERENCES
+
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+Min Joon Seo, Aniruddha Kembhavi, Ali Farhadi, and Hannaneh Hajishirzi. Bidirectional attention flow for machine comprehension. In ICLR, 2017.
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+Yelong Shen, Po-Sen Huang, Jianfeng Gao, and Weizhu Chen. Reasonet: Learning to stop reading in machine comprehension. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1047–1055. ACM, 2017.
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+        "type": "text",
+        "text": "DCN+: MIXED OBJECTIVE AND DEEP RESIDUAL COATTENTION FOR QUESTION ANSWERING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Caiming Xiong∗, Victor Zhong∗, Richard Socher ",
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Salesforce Research   \nPalo Alto, CA 94301, USA   \n{cxiong, vzhong, rsocher}@salesforce.com ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
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+        "text": "Traditional models for question answering optimize using cross entropy loss, which encourages exact answers at the cost of penalizing nearby or overlapping answers that are sometimes equally accurate. We propose a mixed objective that combines cross entropy loss with self-critical policy learning. The objective uses rewards derived from word overlap to solve the misalignment between evaluation metric and optimization objective. In addition to the mixed objective, we improve dynamic coattention networks (DCN) with a deep residual coattention encoder that is inspired by recent work in deep self-attention and residual networks. Our proposals improve model performance across question types and input lengths, especially for long questions that requires the ability to capture long-term dependencies. On the Stanford Question Answering Dataset, our model achieves state-of-the-art results with $7 5 . 1 \\%$ exact match accuracy and $8 3 . 1 \\%$ F1, while the ensemble obtains $7 8 . 9 \\%$ exact match accuracy and $8 6 . 0 \\%$ F1. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Existing state-of-the-art question answering models are trained to produce exact answer spans for a question and a document. In this setting, a ground truth answer used to supervise the model is defined as a start and an end position within the document. Existing training approaches optimize using cross entropy loss over the two positions. However, this suffers from a fundamental disconnect between the optimization, which is tied to the position of a particular ground truth answer span, and the evaluation, which is based on the textual content of the answer. This disconnect is especially harmful in cases where answers that are textually similar to, but distinct in positions from, the ground truth are penalized in the same fashion as answers that are textually dissimilar. For example, suppose we are given the sentence “Some believe that the Golden State Warriors team of 2017 is one of the greatest teams in NBA history”, the question “which team is considered to be one of the greatest teams in NBA history”, and a ground truth answer of “the Golden State Warriors team of 2017”. The span “Warriors” is also a correct answer, but from the perspective of traditional cross entropy based training it is no better than the span “history”. ",
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+        "type": "text",
+        "text": "To address this problem, we propose a mixed objective that combines traditional cross entropy loss over positions with a measure of word overlap trained with reinforcement learning. We obtain the latter objective using self-critical policy learning in which the reward is based on word overlap between the proposed answer and the ground truth answer. Our mixed objective brings two benefits: (i) the reinforcement learning objective encourages answers that are textually similar to the ground truth answer and discourages those that are not; (ii) the cross entropy objective significantly facilitates policy learning by encouraging trajectories that are known to be correct. The resulting objective is one that is both faithful to the evaluation metric and converges quickly in practice. ",
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+        "type": "text",
+        "text": "In addition to our mixed training objective, we extend the Dynamic Coattention Network (DCN) by Xiong et al. (2017) with a deep residual coattention encoder. This allows the network to build richer representations of the input by enabling each input sequence to attend to previous attention contexts. Vaswani et al. (2017) show that the stacking of attention layers helps model long-range dependencies. We merge coattention outputs from each layer by means of residual connections to reduce the length of signal paths. He et al. (2016) show that skip layer connections facilitate signal propagation and alleviate gradient degradation. ",
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+    {
+        "type": "image",
+        "img_path": "images/699cf74e131ca40810b83acce546596d257f708b1b3d1ec693618e6b03cea3d7.jpg",
+        "image_caption": [
+            "Figure 1: Deep residual coattention encoder. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "text": "The combination of the deep residual coattention encoder and the mixed objective leads to higher performance across question types, question lengths, and answer lengths on the Stanford Question Answering Dataset (SQuAD) (Rajpurkar et al., 2016) compared to our DCN baseline. The improvement is especially apparent on long questions, which require the model to capture long-range dependencies between the document and the question. Our model, which we call $\\mathrm { D C N + }$ , achieves state-of-the-art results on SQuAD, with $7 5 . 1 \\%$ exact match accuracy and $8 3 . 1 \\%$ F1. When ensembled, the $\\mathrm { D C N + }$ obtains $78 . 9 \\%$ exact match accuracy and $8 6 . 0 \\%$ F1. ",
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+    {
+        "type": "text",
+        "text": "2 $\\mathrm { D C N + }$ ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            545,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We consider the question answering task in which we are given a document and a question, and are asked to find the answer in the document. Our model is based on the DCN by Xiong et al. (2017), which consists of a coattention encoder and a dynamic decoder. The encoder first encodes the question and the document separately, then builds a codependent representation through coattention. The decoder then produces a start and end point estimate given the coattention. The DCN decoder is dynamic in the sense that it iteratively estimates the start and end positions, stopping when estimates between iterations converge to the same positions or when a predefined maximum number of iterations is reached. We make two significant changes to the DCN by introducing a deep residual coattention encoder and a mixed training objective that combines cross entropy loss from maximum likelihood estimation and reinforcement learning rewards from self-critical policy learning. ",
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+    {
+        "type": "text",
+        "text": "2.1 DEEP RESIDUAL COATTENTION ENCODER ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    {
+        "type": "text",
+        "text": "Because it only has a single-layer coattention encoder, the DCN is limited in its ability to compose complex input representations. Vaswani et al. (2017) proposed stacked self-attention modules to facilitate signal traversal. They also showed that the network’s ability to model long-range dependencies can be improved by reducing the length of signal paths. We propose two modifications to the coattention encoder to leverage these findings. First, we extend the coattention encoder with self-attention by stacking coattention layers. This allows the network to build richer representations over the input. Second, we merge coattention outputs from each layer with residual connections. This reduces the length of signal paths. Our encoder is shown in Figure 1. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Suppose we are given a document of $m$ words and a question of $n$ words. Let $L ^ { D } \\in \\mathbb { R } ^ { e \\times m }$ and $L ^ { \\mathcal { Q } ^ { \\star } } \\in \\mathbb { R } ^ { e \\times n }$ respectively denote the word embeddings for the document and the question, where $e$ is the dimension of the word embeddings. We obtain document encodings $E _ { 1 } ^ { \\hat { D } }$ and question encodings $E _ { 1 } ^ { Q }$ through a bidirectional Long Short-Term Memory Network (LSTM) (Hochreiter & Schmidhuber, 1997), where we use integer subscripts to denote the coattention layer number. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/90b22aaa28c72611cce5ffb215843db98ab405616cbf1220d7a30dcd65740ebb.jpg",
+        "text": "$$\n\\begin{array} { r c l } { E _ { 1 } ^ { D } } & { = } & { \\mathrm { b i L S T M } _ { 1 } \\left( L ^ { D } \\right) \\in \\mathbb { R } ^ { h \\times ( m + 1 ) } } \\\\ { E _ { 1 } ^ { Q } } & { = } & { \\operatorname { t a n h } \\big ( W \\mathrm { \\ b i L S T M } _ { 1 } \\left( L ^ { Q } \\right) + b \\big ) \\in \\mathbb { R } ^ { h \\times ( n + 1 ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, $h$ denotes the hidden state size and the $+ 1$ indicates the presence of an additional sentinel word which allows the coattention to not focus on any part of the input. Like the original DCN, we add a non-linear transform to the question encoding. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We compute the affinity matrix between the document and the question as $A \\ = \\ \\left( E _ { 1 } ^ { D } \\right) ^ { \\boldsymbol { \\mathsf { T } } } E _ { 1 } ^ { Q } \\ \\in$ $\\mathbb { R } ^ { ( m + 1 ) \\times ( n + 1 ) }$ . Let softmax $( X )$ denote the softmax operation over the matrix $X$ that normalizes $X$ column-wise. The document summary vectors and question summary vectors are computed as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/38f5d9209f9169f80956b4cc90f812ad184ef3f6c1e90f6ff562233c6cfb348a.jpg",
+        "text": "$$\n\\begin{array} { r c l } { S _ { 1 } ^ { D } } & { = } & { E _ { 1 } ^ { Q } \\operatorname { s o f t m a x } \\left( A ^ { \\boldsymbol { \\mathsf { T } } } \\right) \\in \\mathbb { R } ^ { h \\times ( m + 1 ) } } \\\\ { S _ { 1 } ^ { Q } } & { = } & { E _ { 1 } ^ { D } \\operatorname { s o f t m a x } \\left( A \\right) \\in \\mathbb { R } ^ { h \\times ( n + 1 ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We define the document coattention context as follows. Note that we drop the dimension corresponding to the sentinel vector – it has already been used during the summary computation and is not a potential position candidate for the decoder. ",
+        "bbox": [
+            174,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1599de97eeda966f30f1083227bbf74011c4d8e247dd98abca219fa173b27cdb.jpg",
+        "text": "$$\n\\begin{array} { r c l } { C _ { 1 } ^ { D } } & { = } & { S _ { 1 } ^ { Q } \\operatorname { s o f t m a x } \\left( A ^ { \\intercal } \\right) \\in \\mathbb { R } ^ { h \\times m } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            449,
+            625,
+            470
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We further encode the summaries using another bidirectional LSTM. ",
+        "bbox": [
+            173,
+            481,
+            624,
+            497
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a29c4cc46e1cf8e1d020d5dc46cf7114b346e870998cd568b4046459283509df.jpg",
+        "text": "$$\n\\begin{array} { r l r } { E _ { 2 } ^ { D } } & { = } & { \\mathrm { b i L S T M _ { 2 } } \\left( S _ { 1 } ^ { D } \\right) \\in \\mathbb { R } ^ { 2 h \\times m } } \\\\ { E _ { 2 } ^ { Q } } & { = } & { \\mathrm { b i L S T M _ { 2 } } \\left( S _ { 1 } ^ { Q } \\right) \\in \\mathbb { R } ^ { 2 h \\times n } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            520,
+            624,
+            571
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Equation 3 to equation 5 describe a single coattention layer. We compute the second coattention layer in a similar fashion. Namely, let coattn denote a multi-valued mapping whose inputs are the two input sequences $E _ { 1 } ^ { D }$ and $E _ { 1 } ^ { Q }$ . We have ",
+        "bbox": [
+            173,
+            580,
+            825,
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/3796b783e7c45f7d0011d2f01ef8030e1fe990bd2081a9a61abddc592edf7e53.jpg",
+        "text": "$$\n{ \\begin{array} { l l l } { \\operatorname { c o a t t n _ { 1 } } \\left( E _ { 1 } ^ { D } , E _ { 1 } ^ { Q } \\right) } & { \\to } & { S _ { 1 } ^ { D } , S _ { 1 } ^ { Q } , C _ { 1 } ^ { D } } \\\\ & & { \\operatorname { c o a t t n _ { 2 } } \\left( E _ { 2 } ^ { D } , E _ { 2 } ^ { Q } \\right) } \\\\ { \\operatorname { c o a t t n _ { 2 } } \\left( E _ { 2 } ^ { D } , E _ { 2 } ^ { Q } \\right) } & { \\to } & { S _ { 2 } ^ { D } , S _ { 2 } ^ { Q } , C _ { 2 } ^ { D } } \\end{array} }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The output of our encoder is then obtained as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c85cbbf39773ce68d9ee1ad4565f25d17e46c288692620c22fbfef1934ab96bb.jpg",
+        "text": "$$\nU = \\mathrm { b i L S T M } \\left( \\mathrm { c o n c a t } \\left( E _ { 1 } ^ { D } ; E _ { 2 } ^ { D } ; S _ { 1 } ^ { D } ; S _ { 2 } ^ { D } ; C _ { 1 } ^ { D } ; C _ { 2 } ^ { D } \\right) \\right) \\in \\mathbb { R } ^ { 2 h \\times m }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where concat $( A , B )$ denotes the concatenation between the matrices $A$ and $B$ along the first dimension. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This encoder is different than the original DCN in its depth and its use of residual connections. We use not only the output of the deep coattention network $\\bar { C } _ { 2 } ^ { D }$ as input to the final bidirectional LSTM, but add skip connections to initial encodings $E _ { 1 } ^ { D }$ , $E _ { 2 } ^ { D }$ 2, summary vectors $S _ { 1 } ^ { D } , S _ { 2 } ^ { D }$ , and coattention context $C _ { 1 } ^ { \\bar { D } }$ . This is akin to transformer networks (Vaswani et al., 2017), which achieved stateof-the-art results on machine translation using deep self-attention layers to help model long-range dependencies, and residual networks (He et al., 2016), which achieved state-of-the-art results in image classification through the addition of skip layer connections to facilitate signal propagation and alleviate gradient degradation. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The DCN produces a distribution over the start position of the answer and a distribution over the end position of the answer. Let $s$ and $e$ denote the respective start and end points of the ground truth answer. Because the decoder of the DCN is dynamic, we denote the start and end distributions produced at the tth decoding step by $p _ { t } ^ { \\mathrm { s t a r t } } \\in \\mathbb { R } ^ { m }$ and $p _ { t } ^ { \\mathrm { e n d } } \\in \\mathbb { R } ^ { m }$ . For convenience, we denote the greedy estimate of the start and end positions by the model at the tth decoding step by $s _ { t }$ and $e _ { t }$ . Moreover, let $\\Theta$ denote the parameters of the model. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Similar to other question answering models, the DCN is supervised using the cross entropy loss on the start position distribution and the end position distribution: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/09da9a027474a38cea7ebc39cd4e15b30d65ef7dfbe39ebbbbddd4f60957c8de.jpg",
+        "text": "$$\nl _ { c e } ( \\Theta ) = - \\sum _ { t } \\left( \\log p _ { t } ^ { \\mathrm { s t a r t } } \\left( s \\mid s _ { t - 1 } , e _ { t - 1 } ; \\Theta \\right) + \\log p _ { t } ^ { \\mathrm { e n d } } \\left( e \\mid s _ { t - 1 } , e _ { t - 1 } ; \\Theta \\right) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Equation 11 states that the model accumulates a cross entropy loss over each position during each decoding step given previous estimates of the start and end positions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The question answering task consists of two evaluation metrics. The first, exact match, is a binary score that denotes whether the answer span produced by the model has exact string match with the ground truth answer span. The second, F1, computes the degree of word overlap between the answer span produced by the model and the ground truth answer span. We note that there is a disconnect between the cross entropy optimization objective and the evaluation metrics. For example, suppose we are given the answer estimates $A$ and $B$ , neither of which match the ground truth positions. However, $A$ has an exact string match with the ground truth answer whereas $B$ does not. The cross entropy objective penalizes $A$ and $B$ equally, despite the former being correct under both evaluation metrics. In the less extreme case where $A$ does not have exact match but has some degree of word overlap with the ground truth, the F1 metric still prefers $A$ over $B$ despite its wrongly predicted positions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We encode this preference using reinforcement learning, using the F1 score as the reward function. Let $\\hat { s _ { t } } \\sim p _ { t } ^ { \\mathrm { s t a r t } }$ and $\\hat { e _ { t } } \\sim p _ { t } ^ { \\mathrm { s t a r t } }$ denote the sampled start and end positions from the estimated distributions at decoding step $t$ . We define a trajectory $\\hat { \\tau }$ as a sequence of sampled start and end points $\\hat { s _ { t } }$ and $\\hat { e _ { t } }$ through all $T$ decoder time steps. The reinforcement learning objective is then the negative expected rewards $R$ over trajectories. ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d2f68ae21de48728324c2f938cd8747de9dea118e406501b5663aa4c79d9a4a7.jpg",
+        "text": "$$\n\\begin{array} { r l r } { l _ { r l } \\left( \\Theta \\right) } & { = } & { - \\mathbb { E } _ { \\hat { r } \\sim p _ { \\tau } } \\left[ R \\left( s , e , \\hat { s } _ { T } , \\hat { e } _ { T } ; \\Theta \\right) \\right] } \\\\ & { \\approx } & { - \\mathbb { E } _ { \\hat { \\tau } \\sim p _ { \\tau } } \\left[ F _ { 1 } \\left( \\mathrm { a n s } \\left( \\hat { \\mathrm { s } } _ { \\mathrm { T } } , \\hat { \\mathrm { e } } _ { \\mathrm { T } } \\right) , \\mathrm { a n s } \\left( \\mathrm { s } , \\mathrm { e } \\right) \\right) - F _ { 1 } \\left( \\mathrm { a n s } \\left( \\mathrm { s } _ { \\mathrm { T } } , \\mathrm { e } _ { \\mathrm { T } } \\right) , \\mathrm { a n s } \\left( \\mathrm { s } , \\mathrm { e } \\right) \\right) \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            223,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We use $F _ { 1 }$ to denote the F1 scoring function and ans $( \\mathrm { s } , \\mathrm { e } )$ to denote the answer span retrieved using the start point $s$ and end point $e$ . In equation 13, instead of using only the F1 word overlap as the reward, we subtract from it a baseline. Greensmith et al. (2001) show that a good baseline reduces the variance of gradient estimates and facilitates convergence. In our case, we employ a self-critic (Konda $\\&$ Tsitsiklis, 1999) that uses the F1 score produced by the current model during greedy inference without teacher forcing. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For ease of notation, we abbreviate $R \\left( s , e , \\hat { s } _ { T } , \\hat { e } _ { T } ; \\Theta \\right)$ as $R$ . As per Sutton et al. (1999) and Schulman et al. (2015), the expected gradient of a non-differentiable reward function can be computed as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/acc5b1ee34f378824fe4560d7d12d5b135bd10fa2871a36cc912f8660e461974.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\displaystyle \\mathcal { T } _ { \\Theta } l _ { r l } ( \\Theta ) } & { = } & { \\displaystyle - \\nabla _ { \\Theta } ( \\mathbb { E } _ { \\hat { r } \\sim p _ { r } } [ R ] ) } \\\\ & { = } & { \\displaystyle - \\mathbb { E } _ { \\hat { r } \\sim p _ { r } } [ R \\nabla _ { \\Theta } \\log p _ { r } ( \\boldsymbol { r } ; \\Theta ) ] } \\\\ & { = } & { \\displaystyle - \\mathbb { E } _ { \\hat { r } \\sim p _ { r } } [ R \\nabla _ { \\Theta } ( \\sum _ { t } ^ { T } ( \\log p _ { t } ^ { \\mathrm { s t a r t } } ( \\hat { s } _ { t } | \\hat { s } _ { t - 1 } , \\hat { e } _ { t - 1 } ; \\Theta ) + \\log p _ { t } ^ { \\mathrm { e n d } } ( \\hat { e } _ { t } | \\hat { s } _ { t - 1 } , \\hat { e } _ { t - 1 } ; \\Theta ) ) ) ] } \\\\ & { \\approx } & { \\displaystyle - R \\nabla _ { \\Theta } ( \\sum _ { t } ^ { T } ( \\log p _ { t } ^ { \\mathrm { s t a r t } } ( \\hat { s } _ { t } | \\hat { s } _ { t - 1 } , \\hat { e } _ { t - 1 } ; \\Theta ) + \\log p _ { t } ^ { \\mathrm { e n d } } ( \\hat { e } _ { t } | \\hat { s } _ { t - 1 } , \\hat { e } _ { t - 1 } ; \\Theta ) ) ) \\qquad ( \\mathrm { l } \\Theta ) \\mathrm { ~ } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "image",
+        "img_path": "images/6004dd53bb1fbb962861f9aee8a80aa6706f83f2e92cdbcdfe5fd35c7b41eefa.jpg",
+        "image_caption": [
+            "Figure 2: Computation of the mixed objective. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "In equation 16, we approximate the expected gradient using a single Monte-Carlo sample $\\tau$ drawn from $p _ { \\tau }$ . This sample trajectory $\\tau$ contains the start and end positions $\\hat { s } _ { t }$ and $\\hat { e } _ { t }$ sampled during all decoding steps. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One of the key problems in applying RL to natural language processing is the discontinuous and discrete space the agent must explore in order to find a good policy. For problems with large exploration space, RL approaches tend to be applied as fine-tuning steps after a maximum likelihood model has already been trained (Paulus et al., 2017; Wu et al., 2016). The resulting model is constrained in its exploration during fine-tuning because it is biased by heavy pretraining. We instead treat the optimization problem as a multi-task learning problem. The first task is to optimize for positional match with the ground truth answer using the the cross entropy objective. The second task is to optimize for word overlap with the ground truth answer with the self-critical reinforcement learning objective. In a similar fashion to Kendall et al. (2017), we combine the two losses using homoscedastic uncertainty as task-dependent weightings. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8e89a40cd3247355c7071fdfd275a64ec27bd891857d6d7a0f53488f0e2af5e7.jpg",
+        "text": "$$\nl = \\frac { 1 } { 2 \\sigma _ { c e } ^ { 2 } } l _ { c e } \\left( \\Theta \\right) + \\frac { 1 } { 2 \\sigma _ { r l } ^ { 2 } } l _ { r l } \\left( \\Theta \\right) + \\log \\sigma _ { c e } ^ { 2 } + \\log \\sigma _ { r l } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, $\\sigma _ { c e }$ and $\\sigma _ { r l }$ are learned parameters. The gradient of the cross entropy objective can be derived using straight-forward backpropagation. The gradient of the self-critical reinforcement learning objective is shown in equation 16. Figure 2 illustrates how the mixed objective is computed. In practice, we find that adding the cross entropy task significantly facilitates policy learning by pruning the space of candidate trajectories - without the former, it is very difficult for policy learning to converge due to the large space of potential answers, documents, and questions. ",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "We train and evaluate our model on the Stanford Question Answering Dataset (SQuAD). We show our test performance of our model against other published models, and demonstrate the importance of our proposals via ablation studies on the development set. To preprocess the corpus, we use the reversible tokenizer from Stanford CoreNLP (Manning et al., 2014). For word embeddings, we use GloVe embeddings pretrained on the 840B Common Crawl corpus (Pennington et al., 2014) as well as character ngram embeddings by Hashimoto et al. (2017). In addition, we concatenate these embeddings with context vectors (CoVe) trained on WMT (McCann et al., 2017). For out of vocabulary words, we set the embeddings and context vectors to zero. We perform word dropout on the document which zeros a word embedding with probability 0.075. In addition, we swap the first maxout layer of the highway maxout network in the DCN decoder with a sparse mixture of experts layer (Shazeer et al., 2017). This layer is similar to the maxout layer, except instead of taking the top scoring expert, we take the top $k = 2$ expert. The model is trained using ADAM (Kingma & Ba, ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2014) with default hyperparameters. Hyperparameters of our model are identical to the DCN. We implement our model using PyTorch. ",
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+    {
+        "type": "table",
+        "img_path": "images/66c06c08b076e38119677fea6f0e73f36f33d9e9c4f51c6f37a1056475e8d87c.jpg",
+        "table_caption": [
+            "3.1 RESULTS "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Single Model Dev</td><td colspan=\"2\">Single Model Test</td><td colspan=\"2\">Ensemble Test</td></tr><tr><td>Model</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td></tr><tr><td>DCN+ (ours)</td><td>74.5%</td><td>83.1%</td><td>75.1%</td><td>83.1%</td><td>78.9%</td><td>86.0%</td></tr><tr><td>rnet</td><td>72.3%</td><td>80.6%</td><td>72.3%</td><td>80.7%</td><td>76.9%</td><td>84.0%</td></tr><tr><td>DCN w/ CoVe (baseline)</td><td>71.3%</td><td>79.9%</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Mnemonic Reader</td><td>70.1%</td><td>79.6%</td><td>69.9%</td><td>79.2%</td><td>73.7%</td><td>81.7%</td></tr><tr><td>Document Reader</td><td>69.5%</td><td>78.8%</td><td>70.0%</td><td>79.0%</td><td>1</td><td>1</td></tr><tr><td>FastQA</td><td>70.3%</td><td>78.5%</td><td>70.8%</td><td>78.9%</td><td>1</td><td>一</td></tr><tr><td>ReasoNet</td><td>1</td><td>1</td><td>69.1%</td><td>78.9%</td><td>73.4%</td><td>81.8%</td></tr><tr><td>SEDT</td><td>67.9%</td><td>77.4%</td><td>68.5%</td><td>78.0%</td><td>73.0%</td><td>80.8%</td></tr><tr><td>BiDAF</td><td>67.7%</td><td>77.3%</td><td>68.0%</td><td>77.3%</td><td>73.7%</td><td>81.5%</td></tr><tr><td>DCN</td><td>65.4%</td><td>75.6%</td><td>66.2%</td><td>75.9%</td><td>71.6%</td><td>80.4%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Table 1: Test performance on SQuAD. The papers are as follows: rnet (Microsoft Asia Natural Language Computing Group, 2017), SEDT (Liu et al., 2017), BiDAF (Seo et al., 2017), DCN w/ CoVe (McCann et al., 2017), ReasoNet (Shen et al., 2017), Document Reader (Chen et al., 2017), FastQA (Weissenborn et al., 2017), DCN (Xiong et al., 2017). The CoVe authors did not submit their model, which we use as our baseline, for SQuAD test evaluation. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "The performance of our model is shown in Table 1. Our model achieves state-of-the-art results on SQuAD dataset with $7 5 . 1 \\%$ exact match accuracy and $8 3 . 1 \\%$ F1. When ensembled, our model obtains $78 . 9 \\%$ exact match accuracy and $8 6 . 0 \\%$ F1. To illustrate the effectiveness of our proposals, we use the DCN with context vectors as a baseline (McCann et al., 2017). This model is identical to the DCN by Xiong et al. (2017), except that it augments the word representations with context vectors trained on WMT16. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Comparison to baseline DCN with CoVe. $\\mathrm { D C N + }$ outperforms the baseline by $3 . 2 \\%$ exact match accuracy and $3 . 2 \\%$ F1 on the SQuAD development set. Figure 3 shows the consistent performance gain of $\\mathrm { D C N + }$ over the baseline across question types, question lengths, and answer lengths. In particular, $\\mathrm { D C N + }$ provides a significant advantage for long questions. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/fbb0659f94d67783dc74da5e903666db4118a5bbafdc6f829825c1345f842277.jpg",
+        "image_caption": [
+            "Figure 3: Performance comparison between $\\mathrm { D C N + }$ and the baseline DCN with CoVe on the SQuAD development set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Ablation study. The contributions of each part of our model are shown in Table 2. We note that the deep residual coattention yielded the highest contribution to model performance, followed by the mixed objective. The sparse mixture of experts layer in the decoder added minor improvements to the model performance. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/4d14784ce7fa3729e1184434916909f1660f9072cdebaf14b4617c974714f90b.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Ablation study on the development set of SQuAD. "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>EM</td><td>△EM</td><td>F1</td><td>△F1</td></tr><tr><td>DCN+ (ours)</td><td>74.5%</td><td>1</td><td>83.1%</td><td>1</td></tr><tr><td>- Deep residual coattention</td><td>73.1%</td><td>-1.4%</td><td>81.5%</td><td>-1.6%</td></tr><tr><td> - Mixed objective</td><td>73.8%</td><td>-0.7%</td><td>82.1%</td><td>-1.0%</td></tr><tr><td>- Mixture of experts</td><td>74.0%</td><td>-0.5%</td><td>82.4%</td><td>-0.7%</td></tr><tr><td>DCN w/ CoVe (baseline)</td><td>71.3%</td><td>-3.2%</td><td>79.9%</td><td>-3.2%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/d28d264cde55c7154152e799bb4c6411d9033cadfa8609e875ca04717a0428f6.jpg",
+        "image_caption": [
+            "Figure 4: Training curve of $\\mathrm { D C N + }$ with and without reinforcement learning. In the latter case, only the cross entropy objective is used. The mixed objective initially performs worse as it begins policy learning from scratch, but quickly outperforms the cross entropy model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Mixed objective convergence. The training curves for $\\mathrm { D C N + }$ with reinforcement learning and $\\mathrm { D C N + }$ without reinforcement learning are shown in Figure 4 to illustrate the effectiveness of our proposed mixed objective. In particular, we note that without mixing in the cross entropy loss, it is extremely difficult to learn the policy. When we combine the cross entropy loss with the reinforcement learning objective, we find that the model initially performs worse early on as it begins policy learning from scratch (shown in Figure 4b). However, with the addition of cross entropy loss, the model quickly learns a reasonable policy and subsequently outperforms the purely cross entropy model (shown in Figure 4a). ",
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+    {
+        "type": "text",
+        "text": "Sample predictions. Figure 5 compares predictions by $\\mathrm { D C N + }$ and by the baseline on the development set. Both models retrieve answers that have sensible entity types. For example, the second example asks for “what game” and both models retrieve an American football game; the third example asks for “type of Turing machine” and both models retrieve a type of turing machine. We find, however, that $\\mathrm { D C N + }$ consistently make less mistakes on finding the correct entity. This is especially apparent in the examples we show, which contain several entities or candidate answers of the correct type. In the first example, Gasquet wrote about the plague and called it “Great Pestilence”. While he likely did think of the plague as a “great pestilence”, the phrase “suggested that it would appear to be some form of ordinary Eastern or bubonic plague” provides evidence for the correct answer – “some form of ordinary Eastern or bubonic plague”. Similarly, the second example states that Thomas Davis was injured in the “NFC Championship Game”, but the game he insisted on playing in is the “Super Bowl”. Finally, “multi-tape” and “single-tape” both appear in the sentence that provides provenance for the answer to the question. However, it is the “single-tape” Turing machine that implies quadratic time. In these examples, $\\mathrm { D C N + }$ finds the correct entity out of ones that have the right type whereas the baseline does not. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            815,
+            341,
+            832
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Neural models for question answering. Current state-of-the-art approaches for question answering over unstructured text tend to be neural approaches. Wang & Jiang (2017) proposed one of the first conditional attention mechanisms in the Match-LSTM encoder. Coattention (Xiong et al., 2017), bidirectional attention flow (Seo et al., 2017), and self-matching attention (Microsoft Asia Natural Language Computing Group, 2017) all build codependent representations of the question and the document. These approaches of conditionally encoding two sequences are widely used in ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "What did Gasquet think the plague was? ",
+        "bbox": [
+            186,
+            99,
+            375,
+            109
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The historian Francis Aidan Gasquet wrote about the 'Great Pestilence' in 1893 and suggested that \"it would appear to be some form of the ordinary Eastern or bubonic plague\". He was able to adopt the epidemiology of the bubonic plague for the Black Death for the second edition in 1908, implicating rats and fleas in the process, and his interpretation was widely accepted for other ancient and medieval epidemics, such as the Justinian plague that was prevalent in the Eastern Roman Empire from 541 to 700 CE. ",
+        "bbox": [
+            186,
+            117,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "What game did Thomas Davis say he would play in, despite breaking a bone earlier on? ",
+        "bbox": [
+            186,
+            195,
+            593,
+            207
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Carolina suffered a major setback when Thomas Davis, an 11-year veteran who had already overcome three ACL tears in his career, went down with a broken arm in the NFC Championship Game. Despite this, he insisted he would still find a way to play in the Super Bowl. His prediction turned out to be accurate. ",
+        "bbox": [
+            187,
+            214,
+            743,
+            244
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "A language solved in quadratic time implies the use of what type of Turing machine? ",
+        "bbox": [
+            186,
+            270,
+            578,
+            280
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language $\\{ \\times \\times | \\times$ is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that \"the time complexities in any two reasonable and general models of computation are polynomially related\" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP. ",
+        "bbox": [
+            186,
+            285,
+            743,
+            354
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 5: Predictions by $\\mathrm { D C N + }$ (red) and DCN with CoVe (blue) on the SQuAD development set. ",
+        "bbox": [
+            178,
+            359,
+            816,
+            375
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "question answering. After building codependent encodings, most models predict the answer by generating the start position and the end position corresponding to the estimated answer span. The generation process utilizes a pointer network (Vinyals et al., 2015) over the positions in the document. Xiong et al. (2017) also introduced the dynamic decoder, which iteratively proposes answers by alternating between start position and end position estimates, and in some cases is able to recover from initially erroneous predictions. ",
+        "bbox": [
+            174,
+            396,
+            825,
+            479
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Neural attention models. Neural attention models saw early adoption in machine translation (Bahdanau et al., 2015) and has since become to de-facto architecture for neural machine translation models. Self-attention, or intra-attention, has been applied to language modeling, sentiment analysis, natural language inference, and abstractive text summarization (Chen et al., 2017; Paulus et al., 2017). Vaswani et al. (2017) extended this idea to a deep self-attentional network which obtained state-of-the-art results in machine translation. Coattention, which builds codependent representations of multiple inputs, has been applied to visual question answering (Lu et al., 2016). Xiong et al. (2017) introduced coattention for question answering. Bidirectional attention flow (Seo et al., 2017) and self-matching attention (Microsoft Asia Natural Language Computing Group, 2017) also build codependent representations between the question and the document. ",
+        "bbox": [
+            174,
+            497,
+            825,
+            636
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning in NLP. Many tasks in natural language processing have evaluation metrics that are not differentiable. Dethlefs & Cuayahuitl ´ (2011) proposed a hierarchical reinforcement learning technique for generating text in a simulated way-finding domain. Narasimhan et al. (2015) applied deep Q-networks to learn policies for text-based games using game rewards as feedback. Li et al. (2016) introduced a neural conversational model trained using policy gradient methods, whose reward function consisted of heuristics for ease of answering, information flow, and semantic coherence. Bahdanau et al. (2017) proposed a general actor-critic temporal-difference method for sequence prediction, performing metric optimization on language modeling and machine translation. Direct word overlap metric optimization has also been applied to summarization (Paulus et al., 2017), and machine translation (Wu et al., 2016). ",
+        "bbox": [
+            174,
+            654,
+            825,
+            792
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            814,
+            318,
+            830
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We introduced $\\mathrm { D C N + }$ , an state-of-the-art question answering model with deep residual coattention trained using a mixed objective that combines cross entropy supervision with self-critical policy learning. We showed that our proposals improve model performance across question types, question lengths, and answer lengths on the Stanford Question Answering Dataset ( SQuAD). On SQuAD, the $\\mathrm { D C N + }$ achieves $7 5 . 1 \\%$ exact match accuracy and $8 3 . 1 \\%$ F1. When ensembled, the $\\mathrm { D C N + }$ obtains $7 8 . 9 \\%$ exact match accuracy and $8 6 . 0 \\%$ F1. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            922
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
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parse/train/HJx7uJStPH/HJx7uJStPH.md

@@ -0,0 +1,300 @@
+# MUSIC SOURCE SEPARATION IN THE WAVEFORM DOMAIN
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Source separation for music is the task of isolating contributions, or stems, from different instruments recorded individually and arranged together to form a song. Such components include voice, bass, drums and any other accompaniments. Contrarily to many audio synthesis tasks where the best performances are achieved by models that directly generate the waveform, the state-of-the-art in source separation for music is to compute masks on the magnitude spectrum. In this paper, we first show that an adaptation of Conv-Tasnet (Luo & Mesgarani, 2019), a waveform-to-waveform model for source separation for speech, significantly beats the state-of-the-art on the MusDB dataset, the standard benchmark of multi-instrument source separation. Second, we observe that Conv-Tasnet follows a masking approach on the input signal, which has the potential drawback of removing parts of the relevant source without the capacity to reconstruct it. We propose Demucs, a new waveform-towaveform model, which has an architecture closer to models for audio generation with more capacity on the decoder. Experiments on the MusDB dataset show that Demucs beats previously reported results in terms of signal to distortion ratio (SDR), but lower than Conv-Tasnet. Human evaluations show that Demucs has significantly higher quality (as assessed by mean opinion score) than Conv-Tasnet, but slightly more contamination from other sources, which explains the difference in SDR. Additional experiments with a larger dataset suggest that the gap in SDR between Demucs and Conv-Tasnet shrinks, showing that our approach is promising.
+
+# 1 INTRODUCTION
+
+Cherry first noticed the “cocktail party effect” (Cherry, 1953): how the human brain is able to separate a single conversation out of a surrounding noise from a room full of people chatting. Bregman later tried to understand how the brain was able to analyse a complex auditory signal and segment it into higher level streams. His framework for auditory scene analysis (Bregman, 1990) spawned its computational counterpart, trying to reproduce or model accomplishments of the brains with algorithmic means (Wang & Brown, 2006), in particular regarding source separation capabilities.
+
+When producing music, recordings of individual instruments called stems are arranged together and mastered into the final song. The goal of source separation is to recover those individual stems from the mixed signal. Unlike the cocktail party problem, there is not a single source of interest to differentiate from an unrelated background noise, but instead a wide variety of tones and timbres playing in a coordinated way. In the SiSec Mus evaluation campaign for music separation (Stöter et al., 2018), those individual stems were grouped into 4 broad categories: (1) drums, (2) bass, (3) other, (4) vocals. Given a music track which is a mixture of these four sources, also called the mix, the goal is to generate four waveforms that correspond to each of the original sources. We consider here the case of supervised source separation, where the training data contain music tracks (i.e., mixtures), together with the ground truth waveform for each of the sources.
+
+State-of-the-art approaches in music source separation still operate on the spectrograms generated by the short-time Fourier transform (STFT). They produce a mask on the magnitude spectrums for each frame and each source, and the output audio is generated by running an inverse STFT on the masked spectrograms reusing the input mixture phase (Takahashi & Mitsufuji, 2017; Takahashi et al., 2018). Several architectures trained end-to-end to directly synthesize the waveforms have been proposed (Lluís et al., 2018; Jansson et al., 2017), but their performances are far below the state-of-the-art: in the last SiSec Mus evaluation campaign (Stöter et al., 2018), the best model that directly predicts waveforms achieves an average signal-to-noise ratio (SDR) over all four sources of 3.2, against 5.3 for the best approach that predicts spectrograms masks (also see Table 1 in Section 6). An upper bound on the performance of all methods relying on masking spectrograms is given by the SDR obtained when using a mask computed using the ground truth sources spectrograms, for instance the Ideal Ratio Mask (IRM) or the Ideal Binary Mask (IBM) oracles. For speech source separation, Luo & Mesgarani (2019) proposed Conv-Tasnet, a model that reuses the masking approach of spectrogram methods but learns the masks jointly with a convolutional front-end, operating directly in the waveform domain for both the inputs and outputs. Conv-Tasnet surpasses both the IRM and IBM oracles.
+
+![](images/c2f8e55ae4154f11a407c116da4b5d8c85e335af72c6f86d64361b47526aad64.jpg)  
+Figure 1: Mel-spectrogram for a 0.8 seconds extract of the bass source from the track “Stich Up” of the MusDB test. From left to right: ground truth, Conv-Tasnet estimate and Demucs estimate. We observe that Conv-Tasnet missed one note entirely.
+
+Our first contribution is to adapt the Conv-Tasnet architecture, originally designed for monophonic speech separation and audio sampled at $8 \mathrm { k H z }$ , to the task of sterephonic music source separation for audio sampled at $4 4 . 1 \mathrm { k H z }$ . Our experiments show that Conv-Tasnet outperforms all previous methods by a large margin, with an SDR of 5.7, but still under the SDR of the IRM oracle at 8.2 (Stöter et al., 2018). However, while Conv-Tasnet separates with a high accuracy the different sources, we observed artifacts when listening to the generated audio: a constant broadband noise, hollow instruments attacks or even missing parts. They are especially noticeable on the drums and bass sources and we give one such example on Figure 1. Conv-Tasnet uses an over-complete linear representation on which it applies a mask obtained from a deep convolutional network. Because both the encoder and decoder are linear, the masking operation cannot synthesize new sounds. We conjecture that the overlap of multiples instruments sometimes lead to a loss of information that is not reversible by a masking operation.
+
+To overcome the limitations of Conv-Tasnet, our second contribution is to propose Demucs, a new architecture for music source separation. Similarly to Conv-Tasnet, Demucs is a deep learning model that directly operates on the raw input waveform and generates a waveform for each source. Demucs is inspired by models for music synthesis rather than masking approaches. It is a U-net architecture with a convolutional encoder and a decoder based on wide transposed convolutions with large strides inspired by recent work on music synthesis (Défossez et al., 2018). The other critical features of the approach are a bidirectional LSTM between the encoder and the decoder, increasing the number of channels exponentially with depth, gated linear units as activation function (Dauphin et al., 2017) which also allow for masking, and a new initialization scheme.
+
+We present experiments on the MusDB benchmark, which first show that both Conv-Tasnet and Demucs achieve performances significantly better than the best methods that operate on the spectrogram, with Conv-Tasnet being better than Demucs in terms of SDR. We also perform human evaluations that compare Conv-Tasnet and our Demucs, which show that Demucs has significantly better perceived quality. The smaller SDR of Demucs is explained by more contamination from other sources. We also conduct an in-depth ablation study of the Demucs architecture to demonstrate the impact of the various design decisions. Finally, we carry out additional experiments by adding 150 songs to the training set. In this experiment, Demucs and TasNet both achieve an SDR of 6.3, suggesting that the gap in terms of SDR between the two models diminishes with more data, making the Demucs approach promising. The 6.3 points of SDR also set a new state-of-the-art, since it improves on the best previous result of 6.0 on the MusDB test set obtained by training with 800 additional songs.
+
+We discuss in more detail the related work in the next Section. We then describe the original ConvTasnet model of Luo & Mesgarani (2018) and its adaptation to music source separation. Our Demucs architecture is detailed in Section 4. We present the experimental protocol in Section 5, and the experimental results compared to the state-of-the-art in Section 6. Finally, we describe the results of the human evaluation and the ablation study.
+
+# 2 RELATED WORK
+
+A first category of methods for supervised music source separation work on time-frequency representations. They predict a power spectrogram for each source and reuse the phase from the input mixture to synthesise individual waveforms. Traditional methods have mostly focused on blind (unsupervised) source separation. Non-negative matrix factorization techniques (Smaragdis et al., 2014) model the power spectrum as a weighted sum of a learnt spectral dictionary, whose elements are grouped into individual sources. Independent component analysis (Hyvärinen et al., 2004) relies on independence assumptions and multiple microphones to separate the sources. Learning a soft/binary mask over power spectrograms has been done using either HMM-based prediction (Roweis, 2001) or segmentation techniques (Bach & Jordan, 2005).
+
+With the development of deep learning, fully supervised methods have gained momentum. Initial work was performed on speech source separation (Grais et al., 2014), followed by works on music using simple fully connected networks over few spectrogram frames (Uhlich et al., 2015), LSTMs (Uhlich et al., 2017), or multi scale convolutional/recurrent networks (Liu & Yang, 2018; Takahashi & Mitsufuji, 2017). Nugraha et al. (2016) showed that Wiener filtering is an efficient post-processing step for spectrogram-based models and it is now used by all top performing models in this category. Those methods have performed the best during the last SiSec 2018 evaluation (Stöter et al., 2018) for source separation on the MusDB (Rafii et al., 2017) dataset. After the evaluation, a reproducible baseline called Open Unmix has been released by Stöter et al. (2019) and matches the top submissions trained only on MusDB. MMDenseLSTM, a model proposed by Takahashi et al. (2018) and trained on 807 unreleased songs currently holds the absolute record of SDR in the SiSec campaign. Both Demucs and Conv-Tasnet obtain significantly higher SDR.
+
+More recently, models operating in the waveform domain have been developed, so far with worse performance than those operating in the spectrogram domain. A convolutional network with a U-Net structure called Wave-U-Net was used first on spectrograms (Jansson et al., 2017) and then adapted to the waveform domain (Stoller et al., 2018). Wave-U-Net was submitted to the SiSec 2018 evaluation campaign with a performance inferior to that of most spectrogram domain models by a large margin. A Wavenet-inspired, although using a regression loss and not auto-regressive, was first used for speech denoising (Rethage et al., 2018) and then adapted to source separation (Lluís et al., 2018). Our model significantly outperforms Wave-U-Net.Given that the Wavenet inspired model performed worse than Wave-U-Net, we did not consider it for comparison.
+
+In the field of monophonic speech source separation, spectrogram masking methods have enjoyed good performance (Kolbæk et al., 2017; Isik et al., 2016). Luo & Mesgarani (2018) developed a waveform domain methods using masking over a learnable front-end obtained from a LSTM that reached the same accuracy. Improvements were obtained by Wang et al. (2018) for spectrogram methods using the unfolding of a few iterations of a phase reconstruction algorithm in the training loss. In the mean time, Luo & Mesgarani (2019) refined their approach, replacing the LSTM with a superposition of dilated convolutions, which improved the SDR and definitely surpassed spectrogram based approaches, including oracles that use the ground truth sources such as the ideal ratio mask (IRM) or the ideal binary mask (IBM). We show in this paper that Conv-Tasnet also outperforms all known methods for music source separation. However it suffers from significantly more artifacts than the Demucs architecture we introduce in this paper, as measured by mean opinion score.
+
+# 3 ADAPTING CONV-TASNET FOR MUSIC SOURCE SEPARATION
+
+We describe in this section the Conv-Tasnet architecture of Luo & Mesgarani (2018) and give the details of how we adapted the architecture to fit the setting of the MusDB dataset.
+
+Overall framework Each source $s$ is represented by a waveform $x _ { s } \in \mathbb { R } ^ { C , T }$ where $C$ is the number of channels (1 for mono, 2 for stereo) and $T$ the number of samples of the waveform. The mixture (i.e., music track) is the sum of all sources $x : = \textstyle \sum _ { s = 1 } ^ { S } x _ { s }$ . We aim at training a model $g$
+
+parameterized by $\theta$ , such that $g ( x ) = ( g _ { s } ( x ; \theta ) ) _ { s = 1 } ^ { S }$ , where $g _ { s } ( x ; \theta )$ is the predicted waveform for source $s$ given $x$ , that minimizes
+
+$$
+\operatorname* { m i n } _ { \theta } \sum _ { x \in \mathcal { D } } \sum _ { s = 1 } ^ { S } L ( g _ { s } ( x ; \theta ) , x _ { s } )
+$$
+
+for some dataset $\mathcal { D }$ and reconstruction error $L$ . The original Conv-Tasnet was trained using a loss called scale-invariant source-to-noise ratio (SI-SNR), similar to the SDR loss described in Section 5. We instead use a simple L1 loss between the estimated and ground truth sources. We discuss in more details regression losses in the context of our Demucs architecture in Section 4.2.
+
+The original Conv-Tasnet architecture Conv-Tasnet (Luo & Mesgarani, 2018) is composed of a learnt front-end that transforms back and forth between the input monophonic mixture waveform sampled at $8 \ \mathrm { k H z }$ and a 128 channels over-complete representation sampled at $1 \ \mathrm { k H z }$ using a convolution as the encoder and a transposed convolution as the decoder, both with a kernel size of 16 and stride of 8. The high dimensional representation is masked through a separation network composed of stacked residual blocks. Each block is composed of a a 1x1 convolution, a PReLU (He et al., 2015b) non linearity, a layer wise normalization over all channels jointly (Ba et al., 2016), a depth-wise separable convolution (Chollet, 2017; Howard et al., 2017) with a kernel size of 3, a stride of 1 and a dilation of $2 ^ { n \bmod N }$ , with $n$ the 0-based index of the block and $N$ an hyper-parameter, and another PReLU and normalization. The output of each block participates to the final mask estimation through a skip connection, preceded by a 1x1 convolution. The original Conv-Tasnet counted $3 \times N$ blocks with $N = 8$ . The mask is obtained summing the output of all blocks and then applying ReLU. The output of the encoder is multiplied by the mask and before going through the decoder.
+
+Conv-Tasnet for music source separation We adapted their architecture to the task of stereophonic music source separation: the original Conv-Tasnet has a receptive field of 1.5 seconds for audio sampled at $8 \ \mathrm { k H z }$ , we take $N = 1 0$ and increased the kernel size (resp. stride) of the encoder/decoder from 16 (resp. 8) to 20 (resp. 10), leading to the same receptive field at $4 4 . 1 \mathrm { k H z }$ . We observed better results using $4 \times N$ blocks instead of $3 \times N$ and 256 channels for the encoder/decoder instead of 128. With those changes, Conv-Tasnet obtained state-of-the-art performance on the MusDB dataset, surpassing all known spectrogram based methods by a large margin as shown in Section 6.
+
+Separating entire songs The original Conv-Tasnet model was designed for short sentences of a few seconds at most. When evaluating it on an entire track, we obtained the best performance by first splitting the input track into chunks of 8 seconds each. We believe this is because of the global layer normalization. During training, only small audio extracts are given, so that a quiet part or a loud part would be scaled back to an average volume. However, when using entire songs as input, it will most likely contain both quiet and loud parts. The normalization will not map both to the same volume, leading to a difference between training and evaluation. We did not observe any side effects when going from one chunk to the next, so we did not look into fancier overlap-add methods.
+
+# 4 THE DEMUCS ARCHITECTURE
+
+The architecture we propose, which we call Demucs, is described in the next few subsections, and the reconstruction loss is discussed in Section 4.2. Demucs takes a stereo mixture as input and outputs a stereo estimate for each source $C = 2$ ). It is an encoder/decoder architecture composed of a convolutional encoder, a bidirectional LSTM, and a convolutional decoder, with the encoder and decoder linked with skip U-Net connections. Similarly to other work in generation in both image (Karras et al., 2018; 2017) and sound (Défossez et al., 2018), we do not use batch normalization (Ioffe & Szegedy, 2015) as our early experiments showed that it was detrimental to the model performance. The overall architecture is depicted in Figure 2a.
+
+# 4.1 CONVOLUTIONAL AUTO-ENCODER
+
+Encoder The encoder is composed of $L : = 6$ stacked convolutional blocks numbered from 1 to $L$ . Each block $i$ is composed of a convolution with kernel size $K : = 8$ , stride $S : = 4$ , $C _ { i - 1 }$ input channels, $C _ { i }$ output channels and ReLU activation, followed by a convolution with kernel size 1, $2 C _ { i }$
+
+(a) Demucs architecture with the mixture waveform as input and the four sources estimates as output. Arrows represents U-Net connections.
+
+![](images/46ac1683040c59307835e69e568b432199f9c653098c5708f11fddff45ad9e98.jpg)  
+Figure 2: Demucs complete architecture on the left, with detailed representation of the encoder and decoder layers on the right.
+
+![](images/0972f1a383778ee80b10f18a7c8accb8a379d028dfd8c7f0012935194dc27c07.jpg)
+
+(b) Detailed view of the layers Decoderi on the top and Encoderi on the bottom. Arrows represent connections to other parts of the model. For convolutions, $C _ { i } n$ (resp $C _ { o } u t )$ is the number of input channels (resp output), $K$ the kernel size and $S$ the stride.
+
+output channels and gated linear units (GLU) as activation function (Dauphin et al., 2017). Since GLUs halve the number of channels, the final output of block $i$ has $C _ { i }$ output channels. A block is described in Figure 2b. Convolutions with kernel width 1 increase the depth and expressivity of the model at low computational cost. As we show in our ablation study 6.2, the usage of GLU activations after these convolutions significantly boost performance. The number of channels in the input mixture is $C _ { 0 } = C = 2$ , while we use $C _ { 1 } : = 1 0 0$ as the number of output channels for the first encoder block. The number of channels is then doubled at each subsequent block, i.e., $C _ { i } : = 2 C _ { i - 1 }$ for $i = 2 . . L$ , so the final number of channels is $C _ { L } = 3 2 0 0$ . We then use a bidirectional LSTM with 2 layers and a hidden size $C _ { L }$ . The LSTM outputs $2 C _ { L }$ channels per time position. We use a linear layer to take that number down to $C _ { L }$ .
+
+Decoder The decoder is mostly the inverse of the encoder. It is composed of $L$ blocks numbered in reverse order from $L$ to 1. The $i$ -th blocks starts with a convolution with stride 1 and kernel width 3 to provide context about adjacent time steps, input/output channels $C _ { i }$ and a ReLU activation. Finally, we use a transposed convolution with kernel width 8 and stride 4, $C _ { i - 1 }$ outputs and ReLU activation. The $S$ sources are synthesized at the final layer only, after all decoder blocks. The final layer is linear with $S \cdot C _ { 0 }$ output channels, one for each source (4 stereo channels in our case), without any additional activation function. Each of these channels directly generate the corresponding waveform.
+
+U-network structure Similarly to Wave-U-Net (Jansson et al., 2017), there are skip connections between the encoder and decoder blocks with the same index, as originally proposed in U-networks (Ronneberger et al., 2015). While the main motivation comes from empirical performances, an advantage of the skip connections is to give a direct access to the original signal, and in particular allows to directly transfers the phase of the input signal to the output, as discussed in Section 4.2. Unlike Wave-U-Net, we use transposed convolutions rather than linear interpolation followed by a convolution with a stride of 1. For the same increase in the receptive field, transposed convolutions require 4 times less operations and memory. This limits the overall number of channels that can be used before running out of memory. As we observed that a large number of channels was key to obtaining good results, we favored the use of transposed convolutions, as explained in Section 6.
+
+Motivation: synthesis vs masking The approach we follow uses the U-Network architecture (Ronneberger et al., 2015; Stoller et al., 2018; Jansson et al., 2017), and builds on transposed convolutions with large number of channels and large strides (4) inspired by the approach to the synthesis of music notes of Défossez et al. (2018). The U-Net skip connections and the gating performed by GLUs imply that this architecture is expressive enough to represent masks on a learnt representation of the input signal, in a similar fashion to Conv-Tasnet. The Demucs approach is then more expressive than Conv-Tasnet, and its main advantages are the multi-scale representations of the input and the non-linear transformations to and from the waveform domain.
+
+# 4.2 LOSS FUNCTION
+
+For the reconstruction loss $L ( g _ { s } ( x ; \theta ) , x _ { s } )$ in equation 1, we either use the average mean square error or average absolute error between waveforms: for a waveform $x _ { s }$ containing $T$ samples and corresponding to source $s$ , a predicted waveform $\hat { x } _ { s }$ and denoting with a subscript $t$ the $t$ -th sample of a waveform, we use one of $L _ { 1 }$ or $L _ { 2 }$ :
+
+$$
+L _ { 1 } ( \boldsymbol { \hat { x } _ { s } } , \boldsymbol { x _ { s } } ) = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } | \hat { x } _ { s , t } - x _ { s , t } | \qquad L _ { 2 } ( \boldsymbol { \hat { x } _ { s } } , \boldsymbol { x _ { s } } ) = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } ( \hat { x } _ { s , t } - x _ { s , t } ) ^ { 2 } .
+$$
+
+In generative models for audio, direct reconstruction losses on waveforms can pose difficulties because they are sensitive to the initial phases of the signals: two signals whose only difference is a shift in the initial phase are perceptually the same, but can have arbitrarily high $L _ { 1 }$ or $L _ { 2 }$ losses. It can be a problem in pure generation tasks because the initial phase of the signal is unknown, and losses on power/magnitude spectrograms are alternative that do not suffer from this lack of specification of the output. Approaches that follow this line either generate spectrograms (e.g., Wang et al., 2017), or use a loss that compares power spectrograms of target/generated waveforms (Défossez et al., 2018).
+
+The problem of invariance to a shift of phase is not as severe in source separation as it is in unconditional generation, because the model has access to the original phase of the signal. The pase can easily be recovered from the skip connections in U-net-style architectures for separation, and is directly used as input of the inverse STFT for methods that generate masks on power spectrograms. As such, losses such as $L 1 / L 2$ are totally valid for source separation. Early experiments with an additional term including the loss of Défossez et al. (2018) did not suggest that it boosts performance, so we did not pursue this direction any further. Most our experiments use $L 1$ loss, and the ablation study presented in Section 6.2 suggests that there is no significant difference between $L 1$ and $L 2$ .
+
+# 4.3 WEIGHT RESCALING AT INITIALIZATION
+
+The initialization of deep neural networks is known to have a critical impact on the overall performances (Glorot & Bengio, 2010; He et al., 2015a), up to the point that Zhang et al. (2019) showed that with a different initialization called fixup, very deep residual networks and transformers can be trained without batch normalization. While Fixup is not designed for U-Net-style skip connections, we observed that the following different initialisation scheme had great positive impact on performances compared to the standard initialization of He et al. (2015a) used in U-Networks.
+
+Considering the so-called Kaiming initialization (He et al., 2015a) as a baseline, let us look at a single convolution layer for which we denote $w$ the weights after the first initialization. We take $\alpha : = \operatorname { s t d } ( w ) / a$ , where $a$ is a reference scale, and replace $w$ by $w ^ { \prime } = w / \sqrt { \alpha }$ . Since the original weights have element-wise order of magnitude $( K C _ { \mathrm { i n } } ) ^ { - 1 / 2 }$ where $K$ is the kernel width and $C _ { \mathrm { i n } }$ the number of output channels, it means that our initialization scheme produces weights of order of magnitude $( K C _ { \mathrm { i n } } ) ^ { - 1 / 4 }$ , together with a non-trivial scale. Based a search over the values [0.01, 0.05, 0.1], we select $a = 0 . 1$ for all the regular and transposed convolutions, see Section 6 for more details. We experimentally observed that on a randomly initialized model applied to an audio extract, it kept the standard deviation of the features along the layers of the same order of magnitude. Without initial rescaling, the output the last layer has a magnitude 20 times smaller than the first.
+
+# 4.4 RANDOMIZED EQUIVARIANT STABILIZATION
+
+A perfect source separation model is time equivariant, i.e. shifting the input mixture by X samples will shift the output Y by the exact same amount. Thanks to its dilated convolutions with a stride of 1, the mask predictor of Conv-Tasnet is naturally time equivariant and even if the encoder/decoder is not strictly equivariant, Conv-Tasnet still verifies this property experimentally (Luo & Mesgarani, 2019). Spectrogram based method will also verify approximately this property. Shifting the input by a small amount will only reflect in the phase of the spectrogram. As the mask is computed only from the magnitude, and the input mixture phase is reused, the output will naturally be shifted by the same amount. On the other hand, we noticed that our architecture did not naturally satisfy this property. We propose a simple workaround called randomized equivariant stabilization, where we sample $S$ random shifts of an input mixture $x$ and average the predictions of our model for each, after having applied the opposite shift. This technique does not require changing the training procedure or network architecture. Using $S = 1 0$ , we obtained a 0.3 SDR gain, see Section 6.2 for more details. It does make evaluation of the model $S$ times slower, however, on a V100 GPU, separating 1 minute of audio at $4 4 . 1 \mathrm { k H z }$ with Demucs takes only 0.8 second. With this technique, separation of 1 minute takes 8 seconds which is still more than 7 times faster than real time.
+
+# 5 EXPERIMENTAL SETUP
+
+# 5.1 EVALUATION FRAMEWORK
+
+MusDB and additional data We use the MusDB dataset (Rafii et al., 2017) , which is composed of 150 songs with full supervision in stereo and sampled at 44100Hz. For each song, we have the exact waveform of the drums, bass, other and vocals parts, i.e. each of the sources. The actual song, the mixture, is the sum of those four parts. The first 84 songs form the train set, the next 16 songs form the valid set (the exact split is defined in the musdb python package) while the remaining 50 are kept for the test set. We collected raw stems for 150 tracks, i.e., individual instrument recordings used in music production software to make a song. We manually assigned each instrument to one of the sources in MusDB. We call this extra supervised data the stem set. We also report the performances of Tasnet and Demucs trained using these 150 songs in addition to the 84 from MusDB, to anaylze the effect of adding more training data.
+
+Source separation metrics Measurements of the performance of source separation models was developed by Vincent et al. for blind source separation (Vincent et al., 2006) and reused for supervised source separation in the SiSec Mus evaluation campaign (Stöter et al., 2018). Similarly to previous work (Stoller et al., 2018; Takahashi & Mitsufuji, 2017; Takahashi et al., 2018), we focus on the SDR (Signal to Distortion Ratio) which measures the log ratio between the volume of the estimated source projection onto the ground truth, and the volume of what is left out of this projection, typically contamination by other sources or artifacts. Other metrics can be defined (SIR and SAR) and we present them in the supplementary material. We used the python package museval which provide a reference implementation for the SiSec Mus 2018 evaluation campaign. As done in the SiSec Mus competition, we report the median over all tracks of the median of the metric over each track computed using the museval package.
+
+# 5.2 BASELINES
+
+As baselines, we selected Open Unmix (Stöter et al., 2019), a 3-layer BiLSTM model with encoding and decoding fully connected layers on spectrogram frames. It was released by the organizers of the SiSec 2018 to act as a strong reproducible baseline and matches the performances of the best candidates trained only on MusDB. We also selected MMDenseLSTM (Takahashi et al., 2018), a multi-band dense net with LSTMs at different scales of the encoder and decoder. This model was submitted as TAK2 and trained with 804 extra labeled songs1. Both MMDenseLSTM and Open Unmix use Wiener filtering (Nugraha et al., 2016) as a last post processing step. The only waveform based method submitted to the evaluation campaign is Wave-U-Net (Stoller et al., 2018) with the identifier STL2. Metrics were downloaded from the SiSec submission repository. for Wave-U-Net and MMDenseLSTM. For Open Unmix they were provided by their authors2. We also provide the metrics for the Ideal Ratio Mask oracle (IRM), which computes the best possible mask using the ground truth sources and is the topline of spectrogram based method (Stöter et al., 2018).
+
+Table 1: Comparison of Conv-Tasnet and Demucs to state-of-the-art models that operate on the waveform (Wave-U-Net) and on spectrograms (Open-Unmix without extra data, MMDenseLSTM with extra data) as well as the IRM oracle on the MusDB test set. The Extra? indicates the number of extra training songs used. We report the median over all tracks of the median SDR over each track, as done in the SiSec Mus evaluation campaign (Stöter et al., 2018). The All column reports the average over all sources. Demucs metrics are averaged over 5 runs, the confidence interval is the standard deviation over $\sqrt { 5 }$ . In bold are the values that are statistically state-of-the-art either with or without extra training data.   
+
+<table><tr><td></td><td></td><td></td><td colspan="5">Test SDR in dB</td></tr><tr><td>Architecture</td><td>Wav?</td><td>Extra?</td><td>All</td><td>Drums</td><td>Bass</td><td>Other</td><td>Vocals</td></tr><tr><td>IRM oracle</td><td>×</td><td>N/A</td><td>8.22</td><td>8.45</td><td>7.12</td><td>7.85</td><td>9.43</td></tr><tr><td>Open-Unmix</td><td>X</td><td>X</td><td>5.33</td><td>5.73</td><td>5.23</td><td>4.02</td><td>6.32</td></tr><tr><td>Wave-U-Net</td><td>√</td><td>X</td><td>3.23</td><td>4.22</td><td>3.21</td><td>2.25</td><td>3.25</td></tr><tr><td>Demucs</td><td>√</td><td>X</td><td>5.58 ±.03</td><td>6.08 ±.06</td><td>5.83 ±.07</td><td>4.12 ±.04</td><td>6.29 ±.07</td></tr><tr><td>Conv-Tasnet</td><td>√</td><td>X</td><td>5.73 ±.03</td><td>6.08 ±.06</td><td>5.66 ±.16</td><td>4.37 ±.02</td><td>6.81 ±.04</td></tr><tr><td>Demucs</td><td></td><td>150</td><td>6.33 ±.02</td><td>7.08 ±.07</td><td>6.70 ±.06</td><td>4.47±.03</td><td>7.05 ±.04</td></tr><tr><td>Conv-Tasnet</td><td>·</td><td>150</td><td>6.32 ±.04</td><td>7.11 ±.13</td><td>7.00 ±.05</td><td>4.44±.03</td><td>6.74 ±.06</td></tr><tr><td>MMDenseLSTM</td><td>X</td><td>804</td><td>6.04</td><td>6.81</td><td>5.40</td><td>4.80</td><td>7.16</td></tr></table>
+
+# 5.3 TRAINING PROCEDURE
+
+Epoch definition and augmentation We define one epoch over the dataset as a pass over all 11-second extracts with a stride of 1 second. We use a random audio shift between 0 and 1 second and keep 10 seconds of audio from there as a training example. We perform the following data augmentation (Uhlich et al., 2017), also used by Open Unmix and MMDenseLSTM: shuffling sources within one batch to generate one new mix, randomly swapping channels. We additionally multiply each source by $\pm 1$ (Nachmani & Wolf, 2019). All Demucs models were trained over 240 epochs. Conv-Tasnet was trained for 360 epochs when trained only on MusDB and 240 when trained with extra data and using only 2-seconds audio extracts.
+
+Training setup and hyper-parameters All models are trained with 16 V100 GPUs with 32GB of RAM. We use a batch size of 64, the Adam (Kingma & Ba, 2015) optimizer with a learning rate was chosen among [3e-4, 5e-4] and the initial number of channels was chosen in [64, 80, 100] based on the L1 loss on the validation set. We obtained best performance with a learning rate of $3 e - 4$ and 100 channels. We then tried 3 different values for the initial weight rescaling reference level described in Section 4.3, [0.01, 0.05, 0.1] and selected 0.1. We computed confidence intervals using 5 random seeds in Table 1. For the ablation study on Table 4, we provide metrics for a single run.
+
+# 6 EXPERIMENTAL RESULTS
+
+In this section, we first provide experimental results on the MusDB dataset for Conv-Tasnet and Demucs compared with state-of-the-art baselines. We then dive into the ablation study of Demucs.
+
+# 6.1 COMPARISON WITH BASELINES
+
+We provide a comparison the state-of-the-art baselines on Table 1. The models on the top half were trained without any extra data while the lower half used unreleased training songs. As no previous work included confidence intervals, we considered the single metric provided by for the baselines as the exact estimate of their mean performance.
+
+Quality of the separation We first observe that Demucs and Conv-Tasnet outperforms all previous methods for music source separation. Conv-Tasnet has significantly higher SDR with 5.73, improving by 0.4 over Open-Unmix. Our proposed Demucs architecture has worse overall performance but matches Conv-Tasnet for the drums source and surpasses it for the bass. When training on 150 extra songs, the two methods have the same overall performance of 6.3 SDR, beating MMDenseLSTM by nearly 0.3 SDR, despite MMDenseLSTM being tained on 804 extra songs. Unlike for speech separation (Luo & Mesgarani, 2019), all methods are still far below the IRM oracle, leaving room for future improvements. We provide results for the other metrics (SIR and SAR) as well as box plots with quantiles over the test set tracks in Appendix B. Audio samples for Demucs, Conv-Tasnet and all baselines are provided in the ICLR link code, with more details given in Appendix A.
+
+Table 2: Mean Opinion Scores (MOS) evaluating the quality and absence of artifacts of the separated audio. 38 people rated 20 samples each, randomly sample from one of the 3 models or the ground truth. There is one sample per track in the MusDB test set and each is 8 seconds long. Ratings of 5 means that the quality is perfect (no artifacts).   
+
+<table><tr><td rowspan="2">Architecture</td><td colspan="5">Quality Mean Opinion Score</td></tr><tr><td>All</td><td>Drums</td><td>Bass</td><td>Other</td><td>Vocals</td></tr><tr><td>Ground truth</td><td>4.46 ±.07</td><td>4.56 ±.13</td><td>4.25 ±.15</td><td>4.45 ±.13</td><td>4.64 ±.13</td></tr><tr><td>Open-Unmix</td><td>3.03 ±.09</td><td>3.10 ±.17</td><td>2.93 ±.20</td><td>3.09 ±0.16</td><td>3.00±.17</td></tr><tr><td>Demucs</td><td>3.22 ±.09</td><td>3.77 ±.15</td><td>3.26 ±.18</td><td>3.32 ±.15</td><td>2.55 ±.20</td></tr><tr><td>Conv-Tasnet</td><td>2.85 ±.08</td><td>3.39 ±.14</td><td>2.29 ±.15</td><td>3.18 ±.14</td><td>2.45 ±.16</td></tr></table>
+
+Table 3: Mean Opinion Scores (MOS) evaluating contamination by other sources. 38 people rated 20 samples each, randomly sampled from one of the 3 models or the ground truth. There is one sample per track in the MusDB test set and each is 8 seconds long. Ratings of 5 means no contamination by other sources.   
+
+<table><tr><td></td><td colspan="5">Contamination Mean Opinion Score</td></tr><tr><td>Architecture</td><td>All</td><td>Drums</td><td>Bass</td><td>Other</td><td>Vocals</td></tr><tr><td>Ground truth</td><td>4.59 ±.07</td><td>4.44 ±.18</td><td>4.69 ±.09</td><td>4.46 ±.13</td><td>4.81 ±.11</td></tr><tr><td>Open-Unmix</td><td>3.27 ±.11</td><td>3.02 ±.19</td><td>4.00 ±.20</td><td>3.11 ±.21</td><td>2.91 ±.20</td></tr><tr><td>Demucs</td><td>3.30 ±.10</td><td>3.08 ±.21</td><td>3.93 ±.18</td><td>3.15 ±.19</td><td>3.02 ±.20</td></tr><tr><td>Conv-Tasnet</td><td>3.42 ±.09</td><td>3.37 ±.17</td><td>3.73 ±.18</td><td>3.46 ±.17</td><td>3.10 ±.17</td></tr></table>
+
+Human evaluations We noticed strong artifacts on the audio separated by Conv-Tasnet, especially for the drums and bass sources: static noise between 1 and $2 \ \mathrm { k H z }$ , hollow instrument attacks or missing notes as illustrated on Figure 1. In order to confirm this observation, we organized a mean opinion score survey. We separated 8 seconds extracts from all of the 50 test set tracks for Conv-Tasnet, Demucs and Open-Unmix. We asked 38 participants to rate 20 samples each, randomly taken from one of the 3 models or the ground truth. For each one, they were required to provide 2 ratings on a scale of 1 to 5. The first one evaluated the quality and absence of artifacts (1: many artifacts and distortion, content is hardly recognizable, 5: perfect quality, no artifacts) and the second one evaluated contamination by other sources (1: contamination if frequent and loud, 5: no contamination). We show the results on Tables 2 and 3. We confirmed that the presence of artifacts in the output of Conv-Tasnet degrades the user experience, with a rating of $2 . 8 5 { \pm } . 0 8 $ against $3 . 2 2 \pm . 0 9$ for Demucs. On the other hand, Conv-Tasnet samples had less contamination by other sources than Open-Unmix or Demucs, although by a small margin, with a rating of $3 . 4 2 \pm . 0 9$ against $3 . 3 0 \pm . 1 0$ for Demucs and $3 . 2 7 \pm . 1 1$ for Open-Unmix.
+
+Training speed We measured the time taken to process a single batch of size 16 with 10 seconds of audio at $4 4 . 1 \mathrm { k H z }$ (the original Wave-U-Net being only trained on $2 2 \mathrm { k H z }$ audio, we double the time for fairness), ignoring data loading and using torch.cuda.synchronize to wait on all kernels to be completed. MMDenseLSTM does not provide a reference implementation. Wave-U-Net takes 1.2 seconds per batch, Open Unmix 0.2 seconds per batch and Demucs 1.6 seconds per batch.
+
+Table 4: Ablation study for the novel elements in our architecture described in Section 4. We use only the train set from MusDB and report best L1 loss over the valid set throughout training as well the SDR on the test set for the epoch that achieved this loss.   
+
+<table><tr><td>Difference</td><td>Valid set L1 loss</td><td>Test set SDR</td></tr><tr><td>no initial weight rescaling</td><td>0.172</td><td>4.94</td></tr><tr><td>no BiLSTM</td><td>0.175</td><td>5.12</td></tr><tr><td>ReLU instead ofGLU</td><td>0.177</td><td>5.19</td></tr><tr><td>no lx1 convolutions in encoder</td><td>0.176</td><td>5.30</td></tr><tr><td>no randomized equivariant stabilization</td><td>N/A</td><td>5.34</td></tr><tr><td>kernel size of 1 in decoder convolutions</td><td>0.166</td><td>5.51</td></tr><tr><td>MSE loss</td><td>N/A</td><td>5.55</td></tr><tr><td>Reference</td><td>0.164</td><td>5.58</td></tr></table>
+
+Conv-Tasnet cannot be trained with such a large sample size, however a single iteration over 2 seconds of audio with a batch size of 4 takes 0.7 seconds.
+
+# 6.2 ABLATION STUDY FOR DEMUCS
+
+We provide an ablation study of the main design decisions for Demucs in Table 4. Given the cost of training a single model, we did not compute confidence intervals for each variation. Yet, any difference inferior to .06, which is the standard deviation observed over 5 repetitions of the Reference model, could be attributed to noise.
+
+We observe a small but not significant improvement when using the L1 loss instead of the MSE loss. Adding a BiLSTM and using the initial weight rescaling described in Section 4.3 provides significant gain, with an extra 0.48 SDR for the first and 0.64 for the second. We observe that using randomized equivariant stabilization as described in Section 4 gives a gain of almost 0.3 SDR. We did not report the validation loss as we only use the stabilization when computing the SDR over the test set. We applied the randomized stabilization to Open-Unmix and Conv-Tasnet with no gain, since, as explained in Section 4.4, both are naturally equivariant with respect to initial time shifts.
+
+We introduced extra convolutions in the encoder and decoder, as described in Sections 4.1. The two proved useful, improving the expressivity of the model, especially when combined with GLU activation. Using a kernel size of 3 instead of 1 in the decoder further improves performance. We conjecture that the context from adjacent time steps helps the output of the transposed convolutions to be consistent through time and reduces potential artifacts arising from using a stride of 4.
+
+# CONCLUSION
+
+We showed that Conv-Tasnet, a state-of-the-art architecture for speech source separation that predicts masks on a learnt front-end over the waveform domain, achieves state-of-the-art performance for music source separation, improving over all previous spectrogram or waveform domain methods by 0.4 SDR. While Conv-Tasnet has excellent performance to separate sources, it suffers from noticeable artifacts as confirmed by human evaluations. We developed an alternative approach, Demucs, that combines the ability to mask over a learnt representation with stronger decoder capacity that allows for audio synthesis. We conjecture that this can be useful when information is lost in the mix of instruments and cannot simply be recovered by masking. We show that our approach produces audio of significantly higher quality as measures by mean opinion scores and matches the SDR of Conv-Tasnet when trained with 150 extra tracks. We believe those results make it a promising alternative to methods based on masking only.
+
+# REFERENCES
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+
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+
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+Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. Technical Report 1710.10196, arXiv, 2017.
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+Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. Technical Report 1812.04948, arXiv, 2018.
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+Morten Kolbæk, Dong Yu, Zheng-Hua Tan, Jesper Jensen, Morten Kolbaek, Dong Yu, Zheng-Hua Tan, and Jesper Jensen. Multitalker speech separation with utterance-level permutation invariant training of deep recurrent neural networks. IEEE/ACM Transactions on Audio, Speech and Language Processing (TASLP), 2017.
+
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+Francesc Lluís, Jordi Pons, and Xavier Serra. End-to-end music source separation: is it possible in the waveform domain? Technical Report 1810.12187, arXiv, 2018.
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+Yi Luo and Nima Mesgarani. Tasnet: time-domain audio separation network for real-time, singlechannel speech separation. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2018.
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+Yi Luo and Nima Mesgarani. Conv-tasnet: Surpassing ideal time–frequency magnitude masking for speech separation. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 2019.
+
+Eliya Nachmani and Lior Wolf. Unsupervised singing voice conversion. Technical Report 1904.06590, arXiv, 2019.
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+Aditya Arie Nugraha, Antoine Liutkus, and Emmanuel Vincent. Multichannel music separation with deep neural networks. In Signal Processing Conference (EUSIPCO), 2016 24th European. IEEE, 2016.
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+Zafar Rafii, Antoine Liutkus, Fabian-Robert Stöter, Stylianos Ioannis Mimilakis, and Rachel Bittner. The musdb18 corpus for music separation, 2017.
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+Dario Rethage, Jordi Pons, and Xavier Serra. A wavenet for speech denoising. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2018.
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+Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computerassisted intervention, 2015.
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+Daniel Stoller, Sebastian Ewert, and Simon Dixon. Wave-u-net: A multi-scale neural network for end-to-end audio source separation. Technical Report 1806.03185, arXiv, 2018.
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+F.-R. Stöter, S. Uhlich, A. Liutkus, and Y. Mitsufuji. Open-unmix - a reference implementation for music source separation. Journal of Open Source Software, 2019.
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+Fabian-Robert Stöter, Antoine Liutkus, and Nobutaka Ito. The 2018 signal separation evaluation campaign. In 14th International Conference on Latent Variable Analysis and Signal Separation, 2018.
+
+Naoya Takahashi and Yuki Mitsufuji. Multi-scale multi-band densenets for audio source separation. In Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA). IEEE, 2017.
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+Naoya Takahashi, Nabarun Goswami, and Yuki Mitsufuji. Mmdenselstm: An efficient combination of convolutional and recurrent neural networks for audio source separation. Technical Report 1805.02410, arXiv, 2018.
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+Stefan Uhlich, Franck Giron, and Yuki Mitsufuji. Deep neural network based instrument extraction from music. In International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2015.
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+Stefan Uhlich, Marcello Porcu, Franck Giron, Michael Enenkl, Thomas Kemp, Naoya Takahashi, and Yuki Mitsufuji. Improving music source separation based on deep neural networks through data augmentation and network blending. In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2017.
+
+Emmanuel Vincent, Rémi Gribonval, and Cédric Févotte. Performance measurement in blind audio source separation. IEEE Transactions on Audio, Speech and Language Processing, 2006.
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+DeLiang Wang and Guy J. Brown (eds.). Computational Auditory Scene Analysis. IEEE Press, Piscataway, NJ, 2006.
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+
+# APPENDIX
+
+# A AUDIO SAMPLES
+
+We provide audio samples taken from the test set of MusDB. They are available through the ICLR code sharing $\mathrm { \ u r l } ^ { 3 }$ along with all the source code to reproduce our experiments. The audio files for the Wave-U-Net and MMDenseLSTM have been obtained from the SiSec Mus 2018 evaluation campaign results website4. For Open Unmix, we generated them from the pretrained UMX model using the reference PyTorch implementation5. We recommend listening to the audio samples with headphones, while being careful with the volume. An HTML page index.html is provided for easier comparison. The following folders are provided:
+
+• Reference: ground truth,   
+• Open Unmix,   
+• WaveUNet,   
+• Demucs: trained only on MusDB,   
+• DemucsExtra: trained on MusDB and an extra 150 songs,   
+• ConvTasnet: trained only on MusDB,   
+• ConvTasnetExtra: trained on MusDB and an extra 150 songs, • MMDenseNetLSTM, trained on MusDB and an extra 804 songs.
+
+# B RESULTS FOR ALL METRICS WITH BOX PLOTS
+
+Reusing the notations from Vincent et al. (2006), let us take a source $j \in { 1 , 2 , 3 , 4 }$ and introduce $P _ { s _ { j } }$ (resp $P _ { \mathbf { s } }$ ) the orthogonal projection on $s _ { j }$ (resp on $\mathrm { S p a n } ( s _ { 1 } , \ldots , s _ { 4 } ) )$ ). We then take with $\hat { s } _ { j }$ the estimate of source $s _ { j }$
+
+$$
+s _ { \mathrm { t a r g e t } } : = P _ { s _ { j } } \big ( \hat { s } _ { j } \big ) \qquad e _ { \mathrm { i n t e r f } } : = P _ { { \tt s } } \big ( \hat { s } _ { j } \big ) - P _ { s _ { j } } \big ( \hat { s } _ { j } \big ) \qquad e _ { \mathrm { a r t i f } } : = \hat { s } _ { j } - P _ { { \tt s } } \big ( \hat { s } _ { j } \big )
+$$
+
+We can now define various signal to noise ratio, expressed in decibels (dB): the source to distortion ratio
+
+$$
+\mathrm { S D R } : = 1 0 \log _ { 1 0 } \frac { \left. s _ { \mathrm { t a r g e t } } \right. ^ { 2 } } { \left. e _ { \mathrm { i n t e r f } } + e _ { \mathrm { a r t i f } } \right. ^ { 2 } } ,
+$$
+
+the source to interference ratio
+
+$$
+\mathrm { S I R } : = 1 0 \log _ { 1 0 } \frac { \left\| s _ { \mathrm { t a r g e t } } \right\| ^ { 2 } } { \left\| e _ { \mathrm { i n t e r f } } \right\| ^ { 2 } }
+$$
+
+and the sources to artifacts ratio
+
+$$
+\mathrm { S A R } : = 1 0 \log _ { 1 0 } \frac { \left\| s _ { \mathrm { t a r g e t } } + e _ { \mathrm { i n t e r f } } \right\| ^ { 2 } } { \left\| e _ { \mathrm { a r t i f } } \right\| ^ { 2 } } .
+$$
+
+As explained in the main paper, extra invariants are added when using the museval package. We refer the reader to Vincent et al. (2006) for more details. We provide box plots for each metric and each target on Figure 3, generated using the notebook provided specifically by the organizers of the SiSec Mus evaluation campaign6. Hereafter, we provide the equivalent of Table 1 in the main paper for both SIR and SAR.
+
+<table><tr><td rowspan="2">Architecture</td><td rowspan="2">Wav?</td><td rowspan="2">Extra?</td><td colspan="6">Test SIR in dB</td></tr><tr><td>All</td><td>Drums</td><td>Bass</td><td>Other</td><td></td><td>Vocals</td></tr><tr><td>IRM oracle</td><td>×</td><td>N/A</td><td>15.53</td><td>15.61</td><td>12.88</td><td>12.84</td><td></td><td>20.78</td></tr><tr><td>Open-Unmix</td><td>X</td><td>X</td><td>10.49</td><td>11.12</td><td>10.93</td><td>6.59</td><td></td><td>13.33</td></tr><tr><td>Wave-U-Net</td><td>√</td><td>X</td><td>6.26</td><td>8.83</td><td>5.78</td><td></td><td>2.37</td><td>8.06</td></tr><tr><td>Demucs</td><td>√</td><td>X</td><td>10.39 ±.07</td><td>11.81 ±.27</td><td>10.55 ±.20</td><td></td><td>5.90 ±.04</td><td>13.31 ±.21</td></tr><tr><td>Conv-Tasnet</td><td>√</td><td>X</td><td>11.47 ±.09</td><td>12.31 ±.09</td><td>11.52 ±.15</td><td></td><td>7.76 ±.07</td><td>14.30 ±.32</td></tr><tr><td>Demucs</td><td></td><td>150</td><td>11.95 ±.09</td><td>13.74 ±.25</td><td>13.03 ±.22</td><td></td><td>7.11 ±.10</td><td>13.94 ±.10</td></tr><tr><td>Conv-Tasnet</td><td>：</td><td>150</td><td>12.24 ±.09</td><td>13.66 ±.14</td><td>13.18 ±.13</td><td></td><td>8.40 ±.08</td><td>13.70 ±.22</td></tr><tr><td>MMDenseLSTM</td><td>×</td><td>804</td><td>12.24</td><td>11.94</td><td>11.59</td><td>8.94</td><td></td><td>16.48</td></tr><tr><td></td><td></td><td colspan="7">Test SAR in dB</td></tr><tr><td>Architecture</td><td>Wav?</td><td>Extra?</td><td>All</td><td>Drums</td><td>Bass</td><td>Other</td><td>Vocals</td><td></td></tr><tr><td>IRM oracle</td><td>×</td><td>N/A</td><td>8.31</td><td>8.40</td><td>7.40</td><td>7.93</td><td>9.51</td><td></td></tr><tr><td>Open-Unmix</td><td></td><td></td><td></td><td></td><td>6.34</td><td>4.74</td><td></td><td></td></tr><tr><td>Wave-U-Net</td><td>X</td><td>X</td><td>5.90 4.49</td><td>6.02 5.29</td><td>4.64</td><td>3.99</td><td>6.52 4.05</td><td></td></tr><tr><td>Demucs</td><td>√</td><td>X</td><td>6.08 ±.01</td><td>6.18 ±.03</td><td>6.41 ±.05</td><td>5.18 ±.06</td><td></td><td>6.54 ±.04</td></tr><tr><td>Conv-Tasnet</td><td>√</td><td>X X</td><td>6.13 ±.04</td><td>6.19 ±.05</td><td>6.60 ±.07</td><td>4.88 ±.02</td><td></td><td>6.87 ±.05</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Demucs</td><td>√</td><td>150</td><td>6.50 ±.02</td><td>7.04 ±.07</td><td>6.68±.04</td><td>5.26 ±.03</td><td></td><td>7.00 ±.05</td></tr><tr><td>Conv-Tasnet</td><td>√</td><td>150</td><td>6.57 ±.02</td><td>7.35 ±.05</td><td>6.96 ±.08</td><td></td><td>4.76 ±.05</td><td>7.20 ±.05</td></tr><tr><td>MMDenseLSTM</td><td>X</td><td>804</td><td>6.50</td><td>6.96</td><td>6.00</td><td>5.55</td><td></td><td>7.48</td></tr></table>
+
+![](images/11355d5566cd6d8a0916a4bfa24cd707a136a1e90497963e4c25c544923500f0.jpg)  
+Figure 3: Boxplot showing the distribution of SDR, SIR and SAR over the tracks of the MusDB test. 16

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+        "type": "text",
+        "text": "MUSIC SOURCE SEPARATION IN THE WAVEFORM DOMAIN ",
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+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        ],
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+        "text": "Source separation for music is the task of isolating contributions, or stems, from different instruments recorded individually and arranged together to form a song. Such components include voice, bass, drums and any other accompaniments. Contrarily to many audio synthesis tasks where the best performances are achieved by models that directly generate the waveform, the state-of-the-art in source separation for music is to compute masks on the magnitude spectrum. In this paper, we first show that an adaptation of Conv-Tasnet (Luo & Mesgarani, 2019), a waveform-to-waveform model for source separation for speech, significantly beats the state-of-the-art on the MusDB dataset, the standard benchmark of multi-instrument source separation. Second, we observe that Conv-Tasnet follows a masking approach on the input signal, which has the potential drawback of removing parts of the relevant source without the capacity to reconstruct it. We propose Demucs, a new waveform-towaveform model, which has an architecture closer to models for audio generation with more capacity on the decoder. Experiments on the MusDB dataset show that Demucs beats previously reported results in terms of signal to distortion ratio (SDR), but lower than Conv-Tasnet. Human evaluations show that Demucs has significantly higher quality (as assessed by mean opinion score) than Conv-Tasnet, but slightly more contamination from other sources, which explains the difference in SDR. Additional experiments with a larger dataset suggest that the gap in SDR between Demucs and Conv-Tasnet shrinks, showing that our approach is promising. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+            336,
+            587
+        ],
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+    },
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+        "text": "Cherry first noticed the “cocktail party effect” (Cherry, 1953): how the human brain is able to separate a single conversation out of a surrounding noise from a room full of people chatting. Bregman later tried to understand how the brain was able to analyse a complex auditory signal and segment it into higher level streams. His framework for auditory scene analysis (Bregman, 1990) spawned its computational counterpart, trying to reproduce or model accomplishments of the brains with algorithmic means (Wang & Brown, 2006), in particular regarding source separation capabilities. ",
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+        "type": "text",
+        "text": "When producing music, recordings of individual instruments called stems are arranged together and mastered into the final song. The goal of source separation is to recover those individual stems from the mixed signal. Unlike the cocktail party problem, there is not a single source of interest to differentiate from an unrelated background noise, but instead a wide variety of tones and timbres playing in a coordinated way. In the SiSec Mus evaluation campaign for music separation (Stöter et al., 2018), those individual stems were grouped into 4 broad categories: (1) drums, (2) bass, (3) other, (4) vocals. Given a music track which is a mixture of these four sources, also called the mix, the goal is to generate four waveforms that correspond to each of the original sources. We consider here the case of supervised source separation, where the training data contain music tracks (i.e., mixtures), together with the ground truth waveform for each of the sources. ",
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+        "type": "text",
+        "text": "State-of-the-art approaches in music source separation still operate on the spectrograms generated by the short-time Fourier transform (STFT). They produce a mask on the magnitude spectrums for each frame and each source, and the output audio is generated by running an inverse STFT on the masked spectrograms reusing the input mixture phase (Takahashi & Mitsufuji, 2017; Takahashi et al., 2018). Several architectures trained end-to-end to directly synthesize the waveforms have been proposed (Lluís et al., 2018; Jansson et al., 2017), but their performances are far below the state-of-the-art: in the last SiSec Mus evaluation campaign (Stöter et al., 2018), the best model that directly predicts waveforms achieves an average signal-to-noise ratio (SDR) over all four sources of 3.2, against 5.3 for the best approach that predicts spectrograms masks (also see Table 1 in Section 6). An upper bound on the performance of all methods relying on masking spectrograms is given by the SDR obtained when using a mask computed using the ground truth sources spectrograms, for instance the Ideal Ratio Mask (IRM) or the Ideal Binary Mask (IBM) oracles. For speech source separation, Luo & Mesgarani (2019) proposed Conv-Tasnet, a model that reuses the masking approach of spectrogram methods but learns the masks jointly with a convolutional front-end, operating directly in the waveform domain for both the inputs and outputs. Conv-Tasnet surpasses both the IRM and IBM oracles. ",
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+        "image_caption": [
+            "Figure 1: Mel-spectrogram for a 0.8 seconds extract of the bass source from the track “Stich Up” of the MusDB test. From left to right: ground truth, Conv-Tasnet estimate and Demucs estimate. We observe that Conv-Tasnet missed one note entirely. "
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+        "text": "Our first contribution is to adapt the Conv-Tasnet architecture, originally designed for monophonic speech separation and audio sampled at $8 \\mathrm { k H z }$ , to the task of sterephonic music source separation for audio sampled at $4 4 . 1 \\mathrm { k H z }$ . Our experiments show that Conv-Tasnet outperforms all previous methods by a large margin, with an SDR of 5.7, but still under the SDR of the IRM oracle at 8.2 (Stöter et al., 2018). However, while Conv-Tasnet separates with a high accuracy the different sources, we observed artifacts when listening to the generated audio: a constant broadband noise, hollow instruments attacks or even missing parts. They are especially noticeable on the drums and bass sources and we give one such example on Figure 1. Conv-Tasnet uses an over-complete linear representation on which it applies a mask obtained from a deep convolutional network. Because both the encoder and decoder are linear, the masking operation cannot synthesize new sounds. We conjecture that the overlap of multiples instruments sometimes lead to a loss of information that is not reversible by a masking operation. ",
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+    {
+        "type": "text",
+        "text": "To overcome the limitations of Conv-Tasnet, our second contribution is to propose Demucs, a new architecture for music source separation. Similarly to Conv-Tasnet, Demucs is a deep learning model that directly operates on the raw input waveform and generates a waveform for each source. Demucs is inspired by models for music synthesis rather than masking approaches. It is a U-net architecture with a convolutional encoder and a decoder based on wide transposed convolutions with large strides inspired by recent work on music synthesis (Défossez et al., 2018). The other critical features of the approach are a bidirectional LSTM between the encoder and the decoder, increasing the number of channels exponentially with depth, gated linear units as activation function (Dauphin et al., 2017) which also allow for masking, and a new initialization scheme. ",
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+        "type": "text",
+        "text": "We present experiments on the MusDB benchmark, which first show that both Conv-Tasnet and Demucs achieve performances significantly better than the best methods that operate on the spectrogram, with Conv-Tasnet being better than Demucs in terms of SDR. We also perform human evaluations that compare Conv-Tasnet and our Demucs, which show that Demucs has significantly better perceived quality. The smaller SDR of Demucs is explained by more contamination from other sources. We also conduct an in-depth ablation study of the Demucs architecture to demonstrate the impact of the various design decisions. Finally, we carry out additional experiments by adding 150 songs to the training set. In this experiment, Demucs and TasNet both achieve an SDR of 6.3, suggesting that the gap in terms of SDR between the two models diminishes with more data, making the Demucs approach promising. The 6.3 points of SDR also set a new state-of-the-art, since it improves on the best previous result of 6.0 on the MusDB test set obtained by training with 800 additional songs. ",
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+        "text": "We discuss in more detail the related work in the next Section. We then describe the original ConvTasnet model of Luo & Mesgarani (2018) and its adaptation to music source separation. Our Demucs architecture is detailed in Section 4. We present the experimental protocol in Section 5, and the experimental results compared to the state-of-the-art in Section 6. Finally, we describe the results of the human evaluation and the ablation study. ",
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+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "A first category of methods for supervised music source separation work on time-frequency representations. They predict a power spectrogram for each source and reuse the phase from the input mixture to synthesise individual waveforms. Traditional methods have mostly focused on blind (unsupervised) source separation. Non-negative matrix factorization techniques (Smaragdis et al., 2014) model the power spectrum as a weighted sum of a learnt spectral dictionary, whose elements are grouped into individual sources. Independent component analysis (Hyvärinen et al., 2004) relies on independence assumptions and multiple microphones to separate the sources. Learning a soft/binary mask over power spectrograms has been done using either HMM-based prediction (Roweis, 2001) or segmentation techniques (Bach & Jordan, 2005). ",
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+        "type": "text",
+        "text": "With the development of deep learning, fully supervised methods have gained momentum. Initial work was performed on speech source separation (Grais et al., 2014), followed by works on music using simple fully connected networks over few spectrogram frames (Uhlich et al., 2015), LSTMs (Uhlich et al., 2017), or multi scale convolutional/recurrent networks (Liu & Yang, 2018; Takahashi & Mitsufuji, 2017). Nugraha et al. (2016) showed that Wiener filtering is an efficient post-processing step for spectrogram-based models and it is now used by all top performing models in this category. Those methods have performed the best during the last SiSec 2018 evaluation (Stöter et al., 2018) for source separation on the MusDB (Rafii et al., 2017) dataset. After the evaluation, a reproducible baseline called Open Unmix has been released by Stöter et al. (2019) and matches the top submissions trained only on MusDB. MMDenseLSTM, a model proposed by Takahashi et al. (2018) and trained on 807 unreleased songs currently holds the absolute record of SDR in the SiSec campaign. Both Demucs and Conv-Tasnet obtain significantly higher SDR. ",
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+    {
+        "type": "text",
+        "text": "More recently, models operating in the waveform domain have been developed, so far with worse performance than those operating in the spectrogram domain. A convolutional network with a U-Net structure called Wave-U-Net was used first on spectrograms (Jansson et al., 2017) and then adapted to the waveform domain (Stoller et al., 2018). Wave-U-Net was submitted to the SiSec 2018 evaluation campaign with a performance inferior to that of most spectrogram domain models by a large margin. A Wavenet-inspired, although using a regression loss and not auto-regressive, was first used for speech denoising (Rethage et al., 2018) and then adapted to source separation (Lluís et al., 2018). Our model significantly outperforms Wave-U-Net.Given that the Wavenet inspired model performed worse than Wave-U-Net, we did not consider it for comparison. ",
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+    {
+        "type": "text",
+        "text": "In the field of monophonic speech source separation, spectrogram masking methods have enjoyed good performance (Kolbæk et al., 2017; Isik et al., 2016). Luo & Mesgarani (2018) developed a waveform domain methods using masking over a learnable front-end obtained from a LSTM that reached the same accuracy. Improvements were obtained by Wang et al. (2018) for spectrogram methods using the unfolding of a few iterations of a phase reconstruction algorithm in the training loss. In the mean time, Luo & Mesgarani (2019) refined their approach, replacing the LSTM with a superposition of dilated convolutions, which improved the SDR and definitely surpassed spectrogram based approaches, including oracles that use the ground truth sources such as the ideal ratio mask (IRM) or the ideal binary mask (IBM). We show in this paper that Conv-Tasnet also outperforms all known methods for music source separation. However it suffers from significantly more artifacts than the Demucs architecture we introduce in this paper, as measured by mean opinion score. ",
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+        "type": "text",
+        "text": "3 ADAPTING CONV-TASNET FOR MUSIC SOURCE SEPARATION ",
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+        "text": "We describe in this section the Conv-Tasnet architecture of Luo & Mesgarani (2018) and give the details of how we adapted the architecture to fit the setting of the MusDB dataset. ",
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+        "text": "Overall framework Each source $s$ is represented by a waveform $x _ { s } \\in \\mathbb { R } ^ { C , T }$ where $C$ is the number of channels (1 for mono, 2 for stereo) and $T$ the number of samples of the waveform. The mixture (i.e., music track) is the sum of all sources $x : = \\textstyle \\sum _ { s = 1 } ^ { S } x _ { s }$ . We aim at training a model $g$ ",
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+        "text": "parameterized by $\\theta$ , such that $g ( x ) = ( g _ { s } ( x ; \\theta ) ) _ { s = 1 } ^ { S }$ , where $g _ { s } ( x ; \\theta )$ is the predicted waveform for source $s$ given $x$ , that minimizes ",
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+    {
+        "type": "equation",
+        "img_path": "images/d970a3dd3d6a4d7418c3de3bb5700960f6eb324be4b824693118ee205bc008b2.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\sum _ { x \\in \\mathcal { D } } \\sum _ { s = 1 } ^ { S } L ( g _ { s } ( x ; \\theta ) , x _ { s } )\n$$",
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+        "type": "text",
+        "text": "for some dataset $\\mathcal { D }$ and reconstruction error $L$ . The original Conv-Tasnet was trained using a loss called scale-invariant source-to-noise ratio (SI-SNR), similar to the SDR loss described in Section 5. We instead use a simple L1 loss between the estimated and ground truth sources. We discuss in more details regression losses in the context of our Demucs architecture in Section 4.2. ",
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+        "type": "text",
+        "text": "The original Conv-Tasnet architecture Conv-Tasnet (Luo & Mesgarani, 2018) is composed of a learnt front-end that transforms back and forth between the input monophonic mixture waveform sampled at $8 \\ \\mathrm { k H z }$ and a 128 channels over-complete representation sampled at $1 \\ \\mathrm { k H z }$ using a convolution as the encoder and a transposed convolution as the decoder, both with a kernel size of 16 and stride of 8. The high dimensional representation is masked through a separation network composed of stacked residual blocks. Each block is composed of a a 1x1 convolution, a PReLU (He et al., 2015b) non linearity, a layer wise normalization over all channels jointly (Ba et al., 2016), a depth-wise separable convolution (Chollet, 2017; Howard et al., 2017) with a kernel size of 3, a stride of 1 and a dilation of $2 ^ { n \\bmod N }$ , with $n$ the 0-based index of the block and $N$ an hyper-parameter, and another PReLU and normalization. The output of each block participates to the final mask estimation through a skip connection, preceded by a 1x1 convolution. The original Conv-Tasnet counted $3 \\times N$ blocks with $N = 8$ . The mask is obtained summing the output of all blocks and then applying ReLU. The output of the encoder is multiplied by the mask and before going through the decoder. ",
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+        "type": "text",
+        "text": "Conv-Tasnet for music source separation We adapted their architecture to the task of stereophonic music source separation: the original Conv-Tasnet has a receptive field of 1.5 seconds for audio sampled at $8 \\ \\mathrm { k H z }$ , we take $N = 1 0$ and increased the kernel size (resp. stride) of the encoder/decoder from 16 (resp. 8) to 20 (resp. 10), leading to the same receptive field at $4 4 . 1 \\mathrm { k H z }$ . We observed better results using $4 \\times N$ blocks instead of $3 \\times N$ and 256 channels for the encoder/decoder instead of 128. With those changes, Conv-Tasnet obtained state-of-the-art performance on the MusDB dataset, surpassing all known spectrogram based methods by a large margin as shown in Section 6. ",
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+        "type": "text",
+        "text": "Separating entire songs The original Conv-Tasnet model was designed for short sentences of a few seconds at most. When evaluating it on an entire track, we obtained the best performance by first splitting the input track into chunks of 8 seconds each. We believe this is because of the global layer normalization. During training, only small audio extracts are given, so that a quiet part or a loud part would be scaled back to an average volume. However, when using entire songs as input, it will most likely contain both quiet and loud parts. The normalization will not map both to the same volume, leading to a difference between training and evaluation. We did not observe any side effects when going from one chunk to the next, so we did not look into fancier overlap-add methods. ",
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+        "type": "text",
+        "text": "4 THE DEMUCS ARCHITECTURE ",
+        "text_level": 1,
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+        "text": "The architecture we propose, which we call Demucs, is described in the next few subsections, and the reconstruction loss is discussed in Section 4.2. Demucs takes a stereo mixture as input and outputs a stereo estimate for each source $C = 2$ ). It is an encoder/decoder architecture composed of a convolutional encoder, a bidirectional LSTM, and a convolutional decoder, with the encoder and decoder linked with skip U-Net connections. Similarly to other work in generation in both image (Karras et al., 2018; 2017) and sound (Défossez et al., 2018), we do not use batch normalization (Ioffe & Szegedy, 2015) as our early experiments showed that it was detrimental to the model performance. The overall architecture is depicted in Figure 2a. ",
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+        "text": "4.1 CONVOLUTIONAL AUTO-ENCODER ",
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+        "text": "Encoder The encoder is composed of $L : = 6$ stacked convolutional blocks numbered from 1 to $L$ . Each block $i$ is composed of a convolution with kernel size $K : = 8$ , stride $S : = 4$ , $C _ { i - 1 }$ input channels, $C _ { i }$ output channels and ReLU activation, followed by a convolution with kernel size 1, $2 C _ { i }$ ",
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+        "type": "text",
+        "text": "(a) Demucs architecture with the mixture waveform as input and the four sources estimates as output. Arrows represents U-Net connections. ",
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+        "image_caption": [
+            "Figure 2: Demucs complete architecture on the left, with detailed representation of the encoder and decoder layers on the right. "
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+        "type": "text",
+        "text": "(b) Detailed view of the layers Decoderi on the top and Encoderi on the bottom. Arrows represent connections to other parts of the model. For convolutions, $C _ { i } n$ (resp $C _ { o } u t )$ is the number of input channels (resp output), $K$ the kernel size and $S$ the stride. ",
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+        "type": "text",
+        "text": "output channels and gated linear units (GLU) as activation function (Dauphin et al., 2017). Since GLUs halve the number of channels, the final output of block $i$ has $C _ { i }$ output channels. A block is described in Figure 2b. Convolutions with kernel width 1 increase the depth and expressivity of the model at low computational cost. As we show in our ablation study 6.2, the usage of GLU activations after these convolutions significantly boost performance. The number of channels in the input mixture is $C _ { 0 } = C = 2$ , while we use $C _ { 1 } : = 1 0 0$ as the number of output channels for the first encoder block. The number of channels is then doubled at each subsequent block, i.e., $C _ { i } : = 2 C _ { i - 1 }$ for $i = 2 . . L$ , so the final number of channels is $C _ { L } = 3 2 0 0$ . We then use a bidirectional LSTM with 2 layers and a hidden size $C _ { L }$ . The LSTM outputs $2 C _ { L }$ channels per time position. We use a linear layer to take that number down to $C _ { L }$ . ",
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+        "type": "text",
+        "text": "Decoder The decoder is mostly the inverse of the encoder. It is composed of $L$ blocks numbered in reverse order from $L$ to 1. The $i$ -th blocks starts with a convolution with stride 1 and kernel width 3 to provide context about adjacent time steps, input/output channels $C _ { i }$ and a ReLU activation. Finally, we use a transposed convolution with kernel width 8 and stride 4, $C _ { i - 1 }$ outputs and ReLU activation. The $S$ sources are synthesized at the final layer only, after all decoder blocks. The final layer is linear with $S \\cdot C _ { 0 }$ output channels, one for each source (4 stereo channels in our case), without any additional activation function. Each of these channels directly generate the corresponding waveform. ",
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+        "type": "text",
+        "text": "U-network structure Similarly to Wave-U-Net (Jansson et al., 2017), there are skip connections between the encoder and decoder blocks with the same index, as originally proposed in U-networks (Ronneberger et al., 2015). While the main motivation comes from empirical performances, an advantage of the skip connections is to give a direct access to the original signal, and in particular allows to directly transfers the phase of the input signal to the output, as discussed in Section 4.2. Unlike Wave-U-Net, we use transposed convolutions rather than linear interpolation followed by a convolution with a stride of 1. For the same increase in the receptive field, transposed convolutions require 4 times less operations and memory. This limits the overall number of channels that can be used before running out of memory. As we observed that a large number of channels was key to obtaining good results, we favored the use of transposed convolutions, as explained in Section 6. ",
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+        "type": "text",
+        "text": "Motivation: synthesis vs masking The approach we follow uses the U-Network architecture (Ronneberger et al., 2015; Stoller et al., 2018; Jansson et al., 2017), and builds on transposed convolutions with large number of channels and large strides (4) inspired by the approach to the synthesis of music notes of Défossez et al. (2018). The U-Net skip connections and the gating performed by GLUs imply that this architecture is expressive enough to represent masks on a learnt representation of the input signal, in a similar fashion to Conv-Tasnet. The Demucs approach is then more expressive than Conv-Tasnet, and its main advantages are the multi-scale representations of the input and the non-linear transformations to and from the waveform domain. ",
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+        "type": "text",
+        "text": "4.2 LOSS FUNCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "For the reconstruction loss $L ( g _ { s } ( x ; \\theta ) , x _ { s } )$ in equation 1, we either use the average mean square error or average absolute error between waveforms: for a waveform $x _ { s }$ containing $T$ samples and corresponding to source $s$ , a predicted waveform $\\hat { x } _ { s }$ and denoting with a subscript $t$ the $t$ -th sample of a waveform, we use one of $L _ { 1 }$ or $L _ { 2 }$ : ",
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+        "img_path": "images/a9b053d73f17bc51a791ffc766670b2c1de44840e2d252cfe1abfa4e7f5e577a.jpg",
+        "text": "$$\nL _ { 1 } ( \\boldsymbol { \\hat { x } _ { s } } , \\boldsymbol { x _ { s } } ) = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } | \\hat { x } _ { s , t } - x _ { s , t } | \\qquad L _ { 2 } ( \\boldsymbol { \\hat { x } _ { s } } , \\boldsymbol { x _ { s } } ) = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } ( \\hat { x } _ { s , t } - x _ { s , t } ) ^ { 2 } .\n$$",
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+        "type": "text",
+        "text": "In generative models for audio, direct reconstruction losses on waveforms can pose difficulties because they are sensitive to the initial phases of the signals: two signals whose only difference is a shift in the initial phase are perceptually the same, but can have arbitrarily high $L _ { 1 }$ or $L _ { 2 }$ losses. It can be a problem in pure generation tasks because the initial phase of the signal is unknown, and losses on power/magnitude spectrograms are alternative that do not suffer from this lack of specification of the output. Approaches that follow this line either generate spectrograms (e.g., Wang et al., 2017), or use a loss that compares power spectrograms of target/generated waveforms (Défossez et al., 2018). ",
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+    },
+    {
+        "type": "text",
+        "text": "The problem of invariance to a shift of phase is not as severe in source separation as it is in unconditional generation, because the model has access to the original phase of the signal. The pase can easily be recovered from the skip connections in U-net-style architectures for separation, and is directly used as input of the inverse STFT for methods that generate masks on power spectrograms. As such, losses such as $L 1 / L 2$ are totally valid for source separation. Early experiments with an additional term including the loss of Défossez et al. (2018) did not suggest that it boosts performance, so we did not pursue this direction any further. Most our experiments use $L 1$ loss, and the ablation study presented in Section 6.2 suggests that there is no significant difference between $L 1$ and $L 2$ . ",
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+    {
+        "type": "text",
+        "text": "4.3 WEIGHT RESCALING AT INITIALIZATION",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The initialization of deep neural networks is known to have a critical impact on the overall performances (Glorot & Bengio, 2010; He et al., 2015a), up to the point that Zhang et al. (2019) showed that with a different initialization called fixup, very deep residual networks and transformers can be trained without batch normalization. While Fixup is not designed for U-Net-style skip connections, we observed that the following different initialisation scheme had great positive impact on performances compared to the standard initialization of He et al. (2015a) used in U-Networks. ",
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+    },
+    {
+        "type": "text",
+        "text": "Considering the so-called Kaiming initialization (He et al., 2015a) as a baseline, let us look at a single convolution layer for which we denote $w$ the weights after the first initialization. We take $\\alpha : = \\operatorname { s t d } ( w ) / a$ , where $a$ is a reference scale, and replace $w$ by $w ^ { \\prime } = w / \\sqrt { \\alpha }$ . Since the original weights have element-wise order of magnitude $( K C _ { \\mathrm { i n } } ) ^ { - 1 / 2 }$ where $K$ is the kernel width and $C _ { \\mathrm { i n } }$ the number of output channels, it means that our initialization scheme produces weights of order of magnitude $( K C _ { \\mathrm { i n } } ) ^ { - 1 / 4 }$ , together with a non-trivial scale. Based a search over the values [0.01, 0.05, 0.1], we select $a = 0 . 1$ for all the regular and transposed convolutions, see Section 6 for more details. We experimentally observed that on a randomly initialized model applied to an audio extract, it kept the standard deviation of the features along the layers of the same order of magnitude. Without initial rescaling, the output the last layer has a magnitude 20 times smaller than the first. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.4 RANDOMIZED EQUIVARIANT STABILIZATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A perfect source separation model is time equivariant, i.e. shifting the input mixture by X samples will shift the output Y by the exact same amount. Thanks to its dilated convolutions with a stride of 1, the mask predictor of Conv-Tasnet is naturally time equivariant and even if the encoder/decoder is not strictly equivariant, Conv-Tasnet still verifies this property experimentally (Luo & Mesgarani, 2019). Spectrogram based method will also verify approximately this property. Shifting the input by a small amount will only reflect in the phase of the spectrogram. As the mask is computed only from the magnitude, and the input mixture phase is reused, the output will naturally be shifted by the same amount. On the other hand, we noticed that our architecture did not naturally satisfy this property. We propose a simple workaround called randomized equivariant stabilization, where we sample $S$ random shifts of an input mixture $x$ and average the predictions of our model for each, after having applied the opposite shift. This technique does not require changing the training procedure or network architecture. Using $S = 1 0$ , we obtained a 0.3 SDR gain, see Section 6.2 for more details. It does make evaluation of the model $S$ times slower, however, on a V100 GPU, separating 1 minute of audio at $4 4 . 1 \\mathrm { k H z }$ with Demucs takes only 0.8 second. With this technique, separation of 1 minute takes 8 seconds which is still more than 7 times faster than real time. ",
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+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTAL SETUP ",
+        "text_level": 1,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 EVALUATION FRAMEWORK ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "MusDB and additional data We use the MusDB dataset (Rafii et al., 2017) , which is composed of 150 songs with full supervision in stereo and sampled at 44100Hz. For each song, we have the exact waveform of the drums, bass, other and vocals parts, i.e. each of the sources. The actual song, the mixture, is the sum of those four parts. The first 84 songs form the train set, the next 16 songs form the valid set (the exact split is defined in the musdb python package) while the remaining 50 are kept for the test set. We collected raw stems for 150 tracks, i.e., individual instrument recordings used in music production software to make a song. We manually assigned each instrument to one of the sources in MusDB. We call this extra supervised data the stem set. We also report the performances of Tasnet and Demucs trained using these 150 songs in addition to the 84 from MusDB, to anaylze the effect of adding more training data. ",
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+    },
+    {
+        "type": "text",
+        "text": "Source separation metrics Measurements of the performance of source separation models was developed by Vincent et al. for blind source separation (Vincent et al., 2006) and reused for supervised source separation in the SiSec Mus evaluation campaign (Stöter et al., 2018). Similarly to previous work (Stoller et al., 2018; Takahashi & Mitsufuji, 2017; Takahashi et al., 2018), we focus on the SDR (Signal to Distortion Ratio) which measures the log ratio between the volume of the estimated source projection onto the ground truth, and the volume of what is left out of this projection, typically contamination by other sources or artifacts. Other metrics can be defined (SIR and SAR) and we present them in the supplementary material. We used the python package museval which provide a reference implementation for the SiSec Mus 2018 evaluation campaign. As done in the SiSec Mus competition, we report the median over all tracks of the median of the metric over each track computed using the museval package. ",
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+    {
+        "type": "text",
+        "text": "5.2 BASELINES",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "As baselines, we selected Open Unmix (Stöter et al., 2019), a 3-layer BiLSTM model with encoding and decoding fully connected layers on spectrogram frames. It was released by the organizers of the SiSec 2018 to act as a strong reproducible baseline and matches the performances of the best candidates trained only on MusDB. We also selected MMDenseLSTM (Takahashi et al., 2018), a multi-band dense net with LSTMs at different scales of the encoder and decoder. This model was submitted as TAK2 and trained with 804 extra labeled songs1. Both MMDenseLSTM and Open Unmix use Wiener filtering (Nugraha et al., 2016) as a last post processing step. The only waveform based method submitted to the evaluation campaign is Wave-U-Net (Stoller et al., 2018) with the identifier STL2. Metrics were downloaded from the SiSec submission repository. for Wave-U-Net and MMDenseLSTM. For Open Unmix they were provided by their authors2. We also provide the metrics for the Ideal Ratio Mask oracle (IRM), which computes the best possible mask using the ground truth sources and is the topline of spectrogram based method (Stöter et al., 2018). ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6e634f3b226a9b3d01dbe6914cd569e9dc613c43ac680e5889caa8eba966c020.jpg",
+        "table_caption": [
+            "Table 1: Comparison of Conv-Tasnet and Demucs to state-of-the-art models that operate on the waveform (Wave-U-Net) and on spectrograms (Open-Unmix without extra data, MMDenseLSTM with extra data) as well as the IRM oracle on the MusDB test set. The Extra? indicates the number of extra training songs used. We report the median over all tracks of the median SDR over each track, as done in the SiSec Mus evaluation campaign (Stöter et al., 2018). The All column reports the average over all sources. Demucs metrics are averaged over 5 runs, the confidence interval is the standard deviation over $\\sqrt { 5 }$ . In bold are the values that are statistically state-of-the-art either with or without extra training data. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td></td><td colspan=\"5\">Test SDR in dB</td></tr><tr><td>Architecture</td><td>Wav?</td><td>Extra?</td><td>All</td><td>Drums</td><td>Bass</td><td>Other</td><td>Vocals</td></tr><tr><td>IRM oracle</td><td>×</td><td>N/A</td><td>8.22</td><td>8.45</td><td>7.12</td><td>7.85</td><td>9.43</td></tr><tr><td>Open-Unmix</td><td>X</td><td>X</td><td>5.33</td><td>5.73</td><td>5.23</td><td>4.02</td><td>6.32</td></tr><tr><td>Wave-U-Net</td><td>√</td><td>X</td><td>3.23</td><td>4.22</td><td>3.21</td><td>2.25</td><td>3.25</td></tr><tr><td>Demucs</td><td>√</td><td>X</td><td>5.58 ±.03</td><td>6.08 ±.06</td><td>5.83 ±.07</td><td>4.12 ±.04</td><td>6.29 ±.07</td></tr><tr><td>Conv-Tasnet</td><td>√</td><td>X</td><td>5.73 ±.03</td><td>6.08 ±.06</td><td>5.66 ±.16</td><td>4.37 ±.02</td><td>6.81 ±.04</td></tr><tr><td>Demucs</td><td></td><td>150</td><td>6.33 ±.02</td><td>7.08 ±.07</td><td>6.70 ±.06</td><td>4.47±.03</td><td>7.05 ±.04</td></tr><tr><td>Conv-Tasnet</td><td>·</td><td>150</td><td>6.32 ±.04</td><td>7.11 ±.13</td><td>7.00 ±.05</td><td>4.44±.03</td><td>6.74 ±.06</td></tr><tr><td>MMDenseLSTM</td><td>X</td><td>804</td><td>6.04</td><td>6.81</td><td>5.40</td><td>4.80</td><td>7.16</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 TRAINING PROCEDURE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Epoch definition and augmentation We define one epoch over the dataset as a pass over all 11-second extracts with a stride of 1 second. We use a random audio shift between 0 and 1 second and keep 10 seconds of audio from there as a training example. We perform the following data augmentation (Uhlich et al., 2017), also used by Open Unmix and MMDenseLSTM: shuffling sources within one batch to generate one new mix, randomly swapping channels. We additionally multiply each source by $\\pm 1$ (Nachmani & Wolf, 2019). All Demucs models were trained over 240 epochs. Conv-Tasnet was trained for 360 epochs when trained only on MusDB and 240 when trained with extra data and using only 2-seconds audio extracts. ",
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+    },
+    {
+        "type": "text",
+        "text": "Training setup and hyper-parameters All models are trained with 16 V100 GPUs with 32GB of RAM. We use a batch size of 64, the Adam (Kingma & Ba, 2015) optimizer with a learning rate was chosen among [3e-4, 5e-4] and the initial number of channels was chosen in [64, 80, 100] based on the L1 loss on the validation set. We obtained best performance with a learning rate of $3 e - 4$ and 100 channels. We then tried 3 different values for the initial weight rescaling reference level described in Section 4.3, [0.01, 0.05, 0.1] and selected 0.1. We computed confidence intervals using 5 random seeds in Table 1. For the ablation study on Table 4, we provide metrics for a single run. ",
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we first provide experimental results on the MusDB dataset for Conv-Tasnet and Demucs compared with state-of-the-art baselines. We then dive into the ablation study of Demucs. ",
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+    {
+        "type": "text",
+        "text": "6.1 COMPARISON WITH BASELINES",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We provide a comparison the state-of-the-art baselines on Table 1. The models on the top half were trained without any extra data while the lower half used unreleased training songs. As no previous work included confidence intervals, we considered the single metric provided by for the baselines as the exact estimate of their mean performance. ",
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+    },
+    {
+        "type": "text",
+        "text": "Quality of the separation We first observe that Demucs and Conv-Tasnet outperforms all previous methods for music source separation. Conv-Tasnet has significantly higher SDR with 5.73, improving by 0.4 over Open-Unmix. Our proposed Demucs architecture has worse overall performance but matches Conv-Tasnet for the drums source and surpasses it for the bass. When training on 150 extra songs, the two methods have the same overall performance of 6.3 SDR, beating MMDenseLSTM by nearly 0.3 SDR, despite MMDenseLSTM being tained on 804 extra songs. Unlike for speech separation (Luo & Mesgarani, 2019), all methods are still far below the IRM oracle, leaving room for future improvements. We provide results for the other metrics (SIR and SAR) as well as box plots with quantiles over the test set tracks in Appendix B. Audio samples for Demucs, Conv-Tasnet and all baselines are provided in the ICLR link code, with more details given in Appendix A. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/a6e71b8dbd587b10efae74f0f4e0c2b960f279557e0647d5172939dbfd73baa8.jpg",
+        "table_caption": [
+            "Table 2: Mean Opinion Scores (MOS) evaluating the quality and absence of artifacts of the separated audio. 38 people rated 20 samples each, randomly sample from one of the 3 models or the ground truth. There is one sample per track in the MusDB test set and each is 8 seconds long. Ratings of 5 means that the quality is perfect (no artifacts). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Architecture</td><td colspan=\"5\">Quality Mean Opinion Score</td></tr><tr><td>All</td><td>Drums</td><td>Bass</td><td>Other</td><td>Vocals</td></tr><tr><td>Ground truth</td><td>4.46 ±.07</td><td>4.56 ±.13</td><td>4.25 ±.15</td><td>4.45 ±.13</td><td>4.64 ±.13</td></tr><tr><td>Open-Unmix</td><td>3.03 ±.09</td><td>3.10 ±.17</td><td>2.93 ±.20</td><td>3.09 ±0.16</td><td>3.00±.17</td></tr><tr><td>Demucs</td><td>3.22 ±.09</td><td>3.77 ±.15</td><td>3.26 ±.18</td><td>3.32 ±.15</td><td>2.55 ±.20</td></tr><tr><td>Conv-Tasnet</td><td>2.85 ±.08</td><td>3.39 ±.14</td><td>2.29 ±.15</td><td>3.18 ±.14</td><td>2.45 ±.16</td></tr></table>",
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/74128a5d3ff2da7181aab9ab020efa5b7a4ea89967566d939ae3c48f3e15247b.jpg",
+        "table_caption": [
+            "Table 3: Mean Opinion Scores (MOS) evaluating contamination by other sources. 38 people rated 20 samples each, randomly sampled from one of the 3 models or the ground truth. There is one sample per track in the MusDB test set and each is 8 seconds long. Ratings of 5 means no contamination by other sources. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"5\">Contamination Mean Opinion Score</td></tr><tr><td>Architecture</td><td>All</td><td>Drums</td><td>Bass</td><td>Other</td><td>Vocals</td></tr><tr><td>Ground truth</td><td>4.59 ±.07</td><td>4.44 ±.18</td><td>4.69 ±.09</td><td>4.46 ±.13</td><td>4.81 ±.11</td></tr><tr><td>Open-Unmix</td><td>3.27 ±.11</td><td>3.02 ±.19</td><td>4.00 ±.20</td><td>3.11 ±.21</td><td>2.91 ±.20</td></tr><tr><td>Demucs</td><td>3.30 ±.10</td><td>3.08 ±.21</td><td>3.93 ±.18</td><td>3.15 ±.19</td><td>3.02 ±.20</td></tr><tr><td>Conv-Tasnet</td><td>3.42 ±.09</td><td>3.37 ±.17</td><td>3.73 ±.18</td><td>3.46 ±.17</td><td>3.10 ±.17</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Human evaluations We noticed strong artifacts on the audio separated by Conv-Tasnet, especially for the drums and bass sources: static noise between 1 and $2 \\ \\mathrm { k H z }$ , hollow instrument attacks or missing notes as illustrated on Figure 1. In order to confirm this observation, we organized a mean opinion score survey. We separated 8 seconds extracts from all of the 50 test set tracks for Conv-Tasnet, Demucs and Open-Unmix. We asked 38 participants to rate 20 samples each, randomly taken from one of the 3 models or the ground truth. For each one, they were required to provide 2 ratings on a scale of 1 to 5. The first one evaluated the quality and absence of artifacts (1: many artifacts and distortion, content is hardly recognizable, 5: perfect quality, no artifacts) and the second one evaluated contamination by other sources (1: contamination if frequent and loud, 5: no contamination). We show the results on Tables 2 and 3. We confirmed that the presence of artifacts in the output of Conv-Tasnet degrades the user experience, with a rating of $2 . 8 5 { \\pm } . 0 8 $ against $3 . 2 2 \\pm . 0 9$ for Demucs. On the other hand, Conv-Tasnet samples had less contamination by other sources than Open-Unmix or Demucs, although by a small margin, with a rating of $3 . 4 2 \\pm . 0 9$ against $3 . 3 0 \\pm . 1 0$ for Demucs and $3 . 2 7 \\pm . 1 1$ for Open-Unmix. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Training speed We measured the time taken to process a single batch of size 16 with 10 seconds of audio at $4 4 . 1 \\mathrm { k H z }$ (the original Wave-U-Net being only trained on $2 2 \\mathrm { k H z }$ audio, we double the time for fairness), ignoring data loading and using torch.cuda.synchronize to wait on all kernels to be completed. MMDenseLSTM does not provide a reference implementation. Wave-U-Net takes 1.2 seconds per batch, Open Unmix 0.2 seconds per batch and Demucs 1.6 seconds per batch. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/880a177f37d3ed511c008d4776632c018a3f0127ab2063bf57082e95cba9f4bb.jpg",
+        "table_caption": [
+            "Table 4: Ablation study for the novel elements in our architecture described in Section 4. We use only the train set from MusDB and report best L1 loss over the valid set throughout training as well the SDR on the test set for the epoch that achieved this loss. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Difference</td><td>Valid set L1 loss</td><td>Test set SDR</td></tr><tr><td>no initial weight rescaling</td><td>0.172</td><td>4.94</td></tr><tr><td>no BiLSTM</td><td>0.175</td><td>5.12</td></tr><tr><td>ReLU instead ofGLU</td><td>0.177</td><td>5.19</td></tr><tr><td>no lx1 convolutions in encoder</td><td>0.176</td><td>5.30</td></tr><tr><td>no randomized equivariant stabilization</td><td>N/A</td><td>5.34</td></tr><tr><td>kernel size of 1 in decoder convolutions</td><td>0.166</td><td>5.51</td></tr><tr><td>MSE loss</td><td>N/A</td><td>5.55</td></tr><tr><td>Reference</td><td>0.164</td><td>5.58</td></tr></table>",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Conv-Tasnet cannot be trained with such a large sample size, however a single iteration over 2 seconds of audio with a batch size of 4 takes 0.7 seconds. ",
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+    {
+        "type": "text",
+        "text": "6.2 ABLATION STUDY FOR DEMUCS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We provide an ablation study of the main design decisions for Demucs in Table 4. Given the cost of training a single model, we did not compute confidence intervals for each variation. Yet, any difference inferior to .06, which is the standard deviation observed over 5 repetitions of the Reference model, could be attributed to noise. ",
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+    {
+        "type": "text",
+        "text": "We observe a small but not significant improvement when using the L1 loss instead of the MSE loss. Adding a BiLSTM and using the initial weight rescaling described in Section 4.3 provides significant gain, with an extra 0.48 SDR for the first and 0.64 for the second. We observe that using randomized equivariant stabilization as described in Section 4 gives a gain of almost 0.3 SDR. We did not report the validation loss as we only use the stabilization when computing the SDR over the test set. We applied the randomized stabilization to Open-Unmix and Conv-Tasnet with no gain, since, as explained in Section 4.4, both are naturally equivariant with respect to initial time shifts. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We introduced extra convolutions in the encoder and decoder, as described in Sections 4.1. The two proved useful, improving the expressivity of the model, especially when combined with GLU activation. Using a kernel size of 3 instead of 1 in the decoder further improves performance. We conjecture that the context from adjacent time steps helps the output of the transposed convolutions to be consistent through time and reduces potential artifacts arising from using a stride of 4. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We showed that Conv-Tasnet, a state-of-the-art architecture for speech source separation that predicts masks on a learnt front-end over the waveform domain, achieves state-of-the-art performance for music source separation, improving over all previous spectrogram or waveform domain methods by 0.4 SDR. While Conv-Tasnet has excellent performance to separate sources, it suffers from noticeable artifacts as confirmed by human evaluations. We developed an alternative approach, Demucs, that combines the ability to mask over a learnt representation with stronger decoder capacity that allows for audio synthesis. We conjecture that this can be useful when information is lost in the mix of instruments and cannot simply be recovered by masking. We show that our approach produces audio of significantly higher quality as measures by mean opinion scores and matches the SDR of Conv-Tasnet when trained with 150 extra tracks. We believe those results make it a promising alternative to methods based on masking only. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "text": "Hongyi Zhang, Yann N Dauphin, and Tengyu Ma. Fixup initialization: Residual learning without normalization. arXiv preprint arXiv:1901.09321, 2019. ",
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            101,
+            281,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A AUDIO SAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            137,
+            349,
+            152
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We provide audio samples taken from the test set of MusDB. They are available through the ICLR code sharing $\\mathrm { \\ u r l } ^ { 3 }$ along with all the source code to reproduce our experiments. The audio files for the Wave-U-Net and MMDenseLSTM have been obtained from the SiSec Mus 2018 evaluation campaign results website4. For Open Unmix, we generated them from the pretrained UMX model using the reference PyTorch implementation5. We recommend listening to the audio samples with headphones, while being careful with the volume. An HTML page index.html is provided for easier comparison. The following folders are provided: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "• Reference: ground truth,   \n• Open Unmix,   \n• WaveUNet,   \n• Demucs: trained only on MusDB,   \n• DemucsExtra: trained on MusDB and an extra 150 songs,   \n• ConvTasnet: trained only on MusDB,   \n• ConvTasnetExtra: trained on MusDB and an extra 150 songs, • MMDenseNetLSTM, trained on MusDB and an extra 804 songs. ",
+        "bbox": [
+            215,
+            281,
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+            441
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B RESULTS FOR ALL METRICS WITH BOX PLOTS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            463,
+            588,
+            479
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Reusing the notations from Vincent et al. (2006), let us take a source $j \\in { 1 , 2 , 3 , 4 }$ and introduce $P _ { s _ { j } }$ (resp $P _ { \\mathbf { s } }$ ) the orthogonal projection on $s _ { j }$ (resp on $\\mathrm { S p a n } ( s _ { 1 } , \\ldots , s _ { 4 } ) )$ ). We then take with $\\hat { s } _ { j }$ the estimate of source $s _ { j }$ ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/57ac880d2f41c4bd004a8e20e210a90c651ebcc6c3dff807878f5bcb1812d6c4.jpg",
+        "text": "$$\ns _ { \\mathrm { t a r g e t } } : = P _ { s _ { j } } \\big ( \\hat { s } _ { j } \\big ) \\qquad e _ { \\mathrm { i n t e r f } } : = P _ { { \\tt s } } \\big ( \\hat { s } _ { j } \\big ) - P _ { s _ { j } } \\big ( \\hat { s } _ { j } \\big ) \\qquad e _ { \\mathrm { a r t i f } } : = \\hat { s } _ { j } - P _ { { \\tt s } } \\big ( \\hat { s } _ { j } \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We can now define various signal to noise ratio, expressed in decibels (dB): the source to distortion ratio ",
+        "bbox": [
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+            573,
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+            601
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ed4b309f1fa335d5d13060520541132c4258d832fbadf3d0e126f508d2199f8.jpg",
+        "text": "$$\n\\mathrm { S D R } : = 1 0 \\log _ { 1 0 } \\frac { \\left. s _ { \\mathrm { t a r g e t } } \\right. ^ { 2 } } { \\left. e _ { \\mathrm { i n t e r f } } + e _ { \\mathrm { a r t i f } } \\right. ^ { 2 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "the source to interference ratio ",
+        "bbox": [
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+            659
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/72369b2c83fa1af85af495371c2ecec07952e4aea7fac2a797cc162339162fea.jpg",
+        "text": "$$\n\\mathrm { S I R } : = 1 0 \\log _ { 1 0 } \\frac { \\left\\| s _ { \\mathrm { t a r g e t } } \\right\\| ^ { 2 } } { \\left\\| e _ { \\mathrm { i n t e r f } } \\right\\| ^ { 2 } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "and the sources to artifacts ratio ",
+        "bbox": [
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+            715
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/51173707194e2b8ef7a64c3a3e5fe369c2136a478a3189a2f33e2914916c72bd.jpg",
+        "text": "$$\n\\mathrm { S A R } : = 1 0 \\log _ { 1 0 } \\frac { \\left\\| s _ { \\mathrm { t a r g e t } } + e _ { \\mathrm { i n t e r f } } \\right\\| ^ { 2 } } { \\left\\| e _ { \\mathrm { a r t i f } } \\right\\| ^ { 2 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            720,
+            625,
+            760
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As explained in the main paper, extra invariants are added when using the museval package. We refer the reader to Vincent et al. (2006) for more details. We provide box plots for each metric and each target on Figure 3, generated using the notebook provided specifically by the organizers of the SiSec Mus evaluation campaign6. Hereafter, we provide the equivalent of Table 1 in the main paper for both SIR and SAR. ",
+        "bbox": [
+            173,
+            775,
+            825,
+            844
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/b2fe405e816ce8f073a01e222a5baeeaa12298150ac0fabf1389b01af8be6f96.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Architecture</td><td rowspan=\"2\">Wav?</td><td rowspan=\"2\">Extra?</td><td colspan=\"6\">Test SIR in dB</td></tr><tr><td>All</td><td>Drums</td><td>Bass</td><td>Other</td><td></td><td>Vocals</td></tr><tr><td>IRM oracle</td><td>×</td><td>N/A</td><td>15.53</td><td>15.61</td><td>12.88</td><td>12.84</td><td></td><td>20.78</td></tr><tr><td>Open-Unmix</td><td>X</td><td>X</td><td>10.49</td><td>11.12</td><td>10.93</td><td>6.59</td><td></td><td>13.33</td></tr><tr><td>Wave-U-Net</td><td>√</td><td>X</td><td>6.26</td><td>8.83</td><td>5.78</td><td></td><td>2.37</td><td>8.06</td></tr><tr><td>Demucs</td><td>√</td><td>X</td><td>10.39 ±.07</td><td>11.81 ±.27</td><td>10.55 ±.20</td><td></td><td>5.90 ±.04</td><td>13.31 ±.21</td></tr><tr><td>Conv-Tasnet</td><td>√</td><td>X</td><td>11.47 ±.09</td><td>12.31 ±.09</td><td>11.52 ±.15</td><td></td><td>7.76 ±.07</td><td>14.30 ±.32</td></tr><tr><td>Demucs</td><td></td><td>150</td><td>11.95 ±.09</td><td>13.74 ±.25</td><td>13.03 ±.22</td><td></td><td>7.11 ±.10</td><td>13.94 ±.10</td></tr><tr><td>Conv-Tasnet</td><td>：</td><td>150</td><td>12.24 ±.09</td><td>13.66 ±.14</td><td>13.18 ±.13</td><td></td><td>8.40 ±.08</td><td>13.70 ±.22</td></tr><tr><td>MMDenseLSTM</td><td>×</td><td>804</td><td>12.24</td><td>11.94</td><td>11.59</td><td>8.94</td><td></td><td>16.48</td></tr><tr><td></td><td></td><td colspan=\"7\">Test SAR in dB</td></tr><tr><td>Architecture</td><td>Wav?</td><td>Extra?</td><td>All</td><td>Drums</td><td>Bass</td><td>Other</td><td>Vocals</td><td></td></tr><tr><td>IRM oracle</td><td>×</td><td>N/A</td><td>8.31</td><td>8.40</td><td>7.40</td><td>7.93</td><td>9.51</td><td></td></tr><tr><td>Open-Unmix</td><td></td><td></td><td></td><td></td><td>6.34</td><td>4.74</td><td></td><td></td></tr><tr><td>Wave-U-Net</td><td>X</td><td>X</td><td>5.90 4.49</td><td>6.02 5.29</td><td>4.64</td><td>3.99</td><td>6.52 4.05</td><td></td></tr><tr><td>Demucs</td><td>√</td><td>X</td><td>6.08 ±.01</td><td>6.18 ±.03</td><td>6.41 ±.05</td><td>5.18 ±.06</td><td></td><td>6.54 ±.04</td></tr><tr><td>Conv-Tasnet</td><td>√</td><td>X X</td><td>6.13 ±.04</td><td>6.19 ±.05</td><td>6.60 ±.07</td><td>4.88 ±.02</td><td></td><td>6.87 ±.05</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Demucs</td><td>√</td><td>150</td><td>6.50 ±.02</td><td>7.04 ±.07</td><td>6.68±.04</td><td>5.26 ±.03</td><td></td><td>7.00 ±.05</td></tr><tr><td>Conv-Tasnet</td><td>√</td><td>150</td><td>6.57 ±.02</td><td>7.35 ±.05</td><td>6.96 ±.08</td><td></td><td>4.76 ±.05</td><td>7.20 ±.05</td></tr><tr><td>MMDenseLSTM</td><td>X</td><td>804</td><td>6.50</td><td>6.96</td><td>6.00</td><td>5.55</td><td></td><td>7.48</td></tr></table>",
+        "bbox": [
+            171,
+            114,
+            864,
+            476
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/11355d5566cd6d8a0916a4bfa24cd707a136a1e90497963e4c25c544923500f0.jpg",
+        "image_caption": [
+            "Figure 3: Boxplot showing the distribution of SDR, SIR and SAR over the tracks of the MusDB test. 16 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            60,
+            818,
+            920
+        ],
+        "page_idx": 15
+    }
+]

parse/train/Hkf2_sC5FX/Hkf2_sC5FX.md

@@ -0,0 +1,494 @@
+# EFFICIENT LIFELONG LEARNING WITH A-GEM
+
+Arslan Chaudhry1, Marc’Aurelio Ranzato2, Marcus Rohrbach2, Mohamed Elhoseiny2 1University of Oxford, 2Facebook AI Research arslan.chaudhry@eng.ox.ac.uk, {ranzato,mrf,elhoseiny}@fb.com
+
+# ABSTRACT
+
+In lifelong learning, the learner is presented with a sequence of tasks, incrementally building a data-driven prior which may be leveraged to speed up learning of a new task. In this work, we investigate the efficiency of current lifelong approaches, in terms of sample complexity, computational and memory cost. Towards this end, we first introduce a new and a more realistic evaluation protocol, whereby learners observe each example only once and hyper-parameter selection is done on a small and disjoint set of tasks, which is not used for the actual learning experience and evaluation. Second, we introduce a new metric measuring how quickly a learner acquires a new skill. Third, we propose an improved version of GEM (Lopez-Paz & Ranzato, 2017), dubbed Averaged GEM (A-GEM), which enjoys the same or even better performance as GEM, while being almost as computationally and memory efficient as EWC (Kirkpatrick et al., 2016) and other regularizationbased methods. Finally, we show that all algorithms including A-GEM can learn even more quickly if they are provided with task descriptors specifying the classification tasks under consideration. Our experiments on several standard lifelong learning benchmarks demonstrate that A-GEM has the best trade-off between accuracy and efficiency.1
+
+# 1 INTRODUCTION
+
+Intelligent systems, whether they are natural or artificial, must be able to quickly adapt to changes in the environment and to quickly learn new skills by leveraging past experiences. While current learning algorithms can achieve excellent performance on a variety of tasks, they strongly rely on copious amounts of supervision in the form of labeled data.
+
+The lifelong learning (LLL) setting attempts at addressing this shortcoming, bringing machine learning closer to a more realistic human learning by acquiring new skills quickly with a small amount of training data, given the experience accumulated in the past. In this setting, the learner is presented with a stream of tasks whose relatedness is not known a priori. The learner has then the potential to learn more quickly a new task, if it can remember how to combine and re-use knowledge acquired while learning related tasks of the past. Of course, for this learning setting to be useful, the model needs to be constrained in terms of amount of compute and memory required. Usually this means that the learner should not be allowed to merely store all examples seen in the past (in which case this reduces the lifelong learning problem to a multitask problem) nor should the learner be engaged in computations that would not be feasible in real-time, as the goal is to quickly learn from a stream of data.
+
+Unfortunately, the established training and evaluation protocol as well as current algorithms for lifelong learning do not satisfy all the above desiderata, namely learning from a stream of data using limited number of samples, limited memory and limited compute. In the most popular training paradigm, the learner does several passes over the data (Kirkpatrick et al., 2016; Aljundi et al., 2018; Rusu et al., 2016; Schwarz et al., 2018), while ideally the model should need only a handful of samples and these should be provided one-by-one in a single pass (Lopez-Paz & Ranzato, 2017). Moreover, when the learner has several hyper-parameters to tune, the current practice is to go over the sequence of tasks several times, each time with a different hyper-parameter value, again ignoring the requirement of learning from a stream of data and, strictly speaking, violating the assumption of the LLL scenario. While some algorithms may work well in a single-pass setting, they unfortunately require a lot of computation (Lopez-Paz & Ranzato, 2017) or their memory scales with the number of tasks (Rusu et al., 2016), which greatly impedes their actual deployment in practical applications.
+
+In this work, we propose an evaluation methodology and an algorithm that better match our desiderata, namely learning efficiently – in terms of training samples, time and memory – from a stream of tasks. First, we propose a new learning paradigm, whereby the learner performs cross validation on a set of tasks which is disjoint from the set of tasks actually used for evaluation (Sec. 2). In this setting, the learner will have to learn and will be tested on an entirely new sequence of tasks and it will perform just a single pass over this data stream. Second, we build upon GEM (Lopez-Paz & Ranzato, 2017), an algorithm which leverages a small episodic memory to perform well in a single pass setting, and propose a small change to the loss function which makes GEM orders of magnitude faster at training time while maintaining similar performance; we dub this variant of GEM, A-GEM (Sec. 4). Third, we explore the use of compositional task descriptors in order to improve the fewshot learning performance within LLL showing that with this additional information the learner can pick up new skills more quickly (Sec. 5). Fourth, we introduce a new metric to measure the speed of learning, which is useful to quantify the ability of a learning algorithm to learn a new task (Sec. 3). And finally, using our new learning paradigm and metric, we demonstrate A-GEM on a variety of benchmarks and against several representative baselines (Sec. 6). Our experiments show that AGEM has a better trade-off between average accuracy and computational/memory cost. Moreover, all algorithms improve their ability to quickly learn a new task when provided with compositional task descriptors, and they do so better and better as they progress through the learning experience.
+
+# 2 LEARNING PROTOCOL
+
+Currently, most works on lifelong learning (Kirkpatrick et al., 2016; Rusu et al., 2016; Shin et al., 2017; Nguyen et al., 2018) adopt a learning protocol which is directly borrowed from supervised learning. There are $T$ tasks, and each task consists of a training, validation and test sets. During training the learner does as many passes over the data of each task as desired. Moreover, hyperparameters are tuned on the validation sets by sweeping over the whole sequence of tasks as many times as required by the cross-validation grid search. Finally, metrics of interest are reported on the test set of each task using the model selected by the previous cross-validation procedure.
+
+Since the current protocol violates our stricter definition of LLL for which the learner can only make a single pass over the data, as we want to emphasize the importance of learning quickly from data, we now introduce a new learning protocol.
+
+We consider two streams of tasks, described by the following ordered sequences of datasets $\mathcal { D } ^ { C V } =$ $\{ \mathcal { D } _ { 1 } , \cdot \cdot \cdot , \mathcal { D } _ { T ^ { c V } } \}$ and $\mathcal { D } ^ { E V } = \{ \mathcal { D } _ { T ^ { C V } + 1 } , \cdot \cdot \cdot , \mathcal { D } _ { T } \}$ , where $\mathcal { D } _ { k } = \{ ( \mathbf { x } _ { i } ^ { \hat { k } } , t _ { i } ^ { k } , y _ { i } ^ { k } ) _ { i = 1 } ^ { n _ { k } } \}$ is the dataset of the $k$ -th task, $T ^ { C V } < T$ (in all our experiments $T ^ { C V } = 3$ while $T = 2 0$ ), and we assume that all datasets are drawn from the same distribution over tasks. To avoid cluttering of the notation, we let the context specify whether $\mathcal { D } _ { k }$ refers to the training or test set of the $k$ -th dataset.
+
+${ \mathcal { D } } ^ { C V }$ is the stream of datasets which will be used during cross-validation; ${ \mathcal { D } } ^ { C V }$ allows the learner to replay all samples multiple times for the purposes of model hyper-parameter selection. Instead, $\mathcal { D } ^ { E V }$ is the actual dataset used for final training and evaluation on the test set; the learner will observe training examples from $\mathcal { D } ^ { E V }$ once and only once, and all metrics will be reported on the test sets of $\mathcal { D } ^ { E \widetilde { V } }$ . Since the regularization-based approaches for lifelong learning (Kirkpatrick et al., 2016; Zenke et al., 2017) are rather sensitive to the choice of the regularization hyper-parameter, we introduced the set $\mathcal { D } ^ { C V }$ , as it seems reasonable in practical applications to have similar tasks that can be used for tuning the system. However, the actual training and testing are then performed on $\mathcal { D } ^ { E V }$ using a single pass over the data. See Algorithm 1 for a summary of the training and evaluation protocol.
+
+Each example in any of these dataset consists of a triplet defined by an input $( \mathbf { x } ^ { k } \in \mathcal { X } )$ , task descriptor ${ \mathbf { } } ^ { t ^ { k } } \in { \mathcal { T } }$ , see Sec. 5 for examples) and a target vector $( y ^ { k } \in \mathbf { y } ^ { k } )$ , where $\mathbf { y } ^ { k }$ is the set of labels specific to task $k$ and $\mathbf { y } ^ { k } \subset \mathcal { y }$ . While observing the data, the goal is to learn a predictor $f _ { \theta } : \mathcal { X } \times \mathcal { T }  \mathcal { Y }$ , parameterized by $\theta \in \mathbb { R } ^ { P }$ (a neural network in our case), that can map any test pair $\mathbf { \Psi } ( \mathbf { x } , t )$ to a target $y$ .
+
+Algorithm 1 Learning and Evaluation Protocols   
+1: for $h$ in hyper-parameter list do . Cross-validation loop, executing multiple passes over $\mathcal { D } ^ { C V }$   
+2: for $k = 1$ to $T ^ { C V }$ do . Learn over data stream $\mathcal { D } ^ { C V }$ using $h$   
+3: for $i = 1$ to $n _ { k }$ do $\triangleright$ Single pass over $\mathcal { D } _ { k }$   
+4: Update $f _ { \theta }$ using $( \mathbf { x } _ { i } ^ { k } , t _ { i } ^ { k } , y _ { i } ^ { k } )$ and hyper-parameter $h$   
+5: Update metrics on test set of $\mathcal { D } ^ { C V }$   
+6: end for   
+7: end for   
+8: end for   
+9: Select best hyper-parameter setting, $h ^ { * }$ , based on average accuracy of test set of $\mathcal { D } ^ { C V }$ , see Eq. 1.   
+10: Reset $f _ { \theta }$ .   
+11: Reset all metrics.   
+12: for $k = T ^ { C V } + 1$ to $T$ do . Actual learning over datastream DEV   
+13: for $i = 1$ to $n _ { k }$ do $\triangleright$ Single pass over $\mathcal { D } _ { k }$   
+14: Update $f _ { \theta }$ using $( \mathbf { x } _ { i } ^ { k } , t _ { i } ^ { k } , y _ { i } ^ { k } )$ and hyper-parameter $h ^ { * }$   
+15: Update metrics on test set of $\mathcal { D } ^ { E V }$   
+16: end for   
+17: end for   
+18: Report metrics on test set of $\mathcal { D } ^ { E V }$ .
+
+# 3 METRICS
+
+Below we describe the metrics used to evaluate the LLL methods studied in this work. In addition to Average Accuracy $( A )$ and Forgetting Measure $( F )$ (Chaudhry et al., 2018), we define a new measure, the Learning Curve Area (LCA), that captures how quickly a model learns.
+
+The training dataset of each task, $\mathcal { D } _ { k }$ , consists of a total $B _ { k }$ mini-batches. After each presentation of a mini-batch of task $k$ , we evaluate the performance of the learner on all the tasks using the corresponding test sets. Let $a _ { k , i , j } \in [ 0 , 1 ]$ be the accuracy evaluated on the test set of task $j$ , after the model has been trained with the $i$ -th mini-batch of task $k$ . Assuming the first learning task in the continuum is indexed by 1 (it will be $T ^ { C V } + 1$ for $\mathcal { D } ^ { E V }$ ) and the last one by $T$ (it will be $T ^ { C V }$ for $\mathcal { D } ^ { C V } .$ ), we define the following metrics:
+
+Average Accuracy $( A \in [ 0 , 1 ] )$ ) Average accuracy after the model has been trained continually with all the mini-batches up till task $k$ is defined as:
+
+$$
+A _ { k } = \frac { 1 } { k } \sum _ { j = 1 } ^ { k } a _ { k , B _ { k } , j }
+$$
+
+In particular, $A _ { T }$ is the average accuracy on all the tasks after the last task has been learned; this is the most commonly used metric used in LLL.
+
+Forgetting Measure $( F \in [ - 1 , 1 ] )$ ) (Chaudhry et al., 2018) Average forgetting after the model has been trained continually with all the mini-batches up till task $k$ is defined as:
+
+$$
+F _ { k } = { \frac { 1 } { k - 1 } } \sum _ { j = 1 } ^ { k - 1 } f _ { j } ^ { k }
+$$
+
+where $f _ { j } ^ { k }$ is the forgetting on task $\cdot _ { j } ,$ after the model is trained with all the mini-batches up till task $k$ and computed as:
+
+$$
+f _ { j } ^ { k } = \operatorname* { m a x } _ { l \in \{ 1 , \cdots , k - 1 \} } a _ { l , B _ { l } , j } - a _ { k , B _ { k } , j }
+$$
+
+Measuring forgetting after all tasks have been learned is important for a two-fold reason. It quantifies the accuracy drop on past tasks, and it gives an indirect notion of how quickly a model may learn a new task, since a forgetful model will have little knowledge left to transfer, particularly so if the new task relates more closely to one of the very first tasks encountered during the learning experience.
+
+Learning Curve Area $( \mathbf { L C A } \in [ 0 , 1 ] )$ ) Let us first define an average $b$ -shot performance (where $b$ is the mini-batch number) after the model has been trained for all the $T$ tasks as:
+
+$$
+Z _ { b } = \frac { 1 } { T } \sum _ { k = 1 } ^ { T } a _ { k , b , k }
+$$
+
+LCA at $\beta$ is the area of the convergence curve $Z _ { b }$ as a function of $b \in [ 0 , \beta ]$ :
+
+$$
+\mathrm { L C A } _ { \beta } = \frac { 1 } { \beta + 1 } \int _ { 0 } ^ { \beta } Z _ { b } d b = \frac { 1 } { \beta + 1 } \sum _ { b = 0 } ^ { \beta } Z _ { b }
+$$
+
+LCA has an intuitive interpretation. $\mathrm { L C A _ { 0 } }$ is the average 0-shot performance, the same as forward transfer in Lopez-Paz & Ranzato (2017). $\mathrm { L C A } _ { \beta }$ is the area under the $Z _ { b }$ curve, which is high if the 0-shot performance is good and if the learner learns quickly. In particular, there could be two models with the same $Z _ { \beta }$ or $A _ { T }$ , but very different $\mathrm { L C A } _ { \beta }$ because one learns much faster than the other while they both eventually obtain the same final accuracy. This metric aims at discriminating between these two cases, and it makes sense for relatively small values of $\beta$ since we are interested in models that learn from few examples.
+
+# 4 AVERAGED GRADIENT EPISODIC MEMORY (A-GEM)
+
+So far we discussed a better training and evaluation protocol for LLL and a new metric to measure the speed of learning. Next, we review GEM (Lopez-Paz & Ranzato, 2017), which is an algorithm that has been shown to work well in the single epoch setting. Unfortunately, GEM is very intensive in terms of computational and memory cost, which motivates our efficient variant, dubbed A-GEM. In Sec. 5, we will describe how compositional task descriptors can be leveraged to further speed up learning in the few shot regime.
+
+GEM avoids catastrophic forgetting by storing an episodic memory $\mathcal { M } _ { k }$ for each task $k$ . While minimizing the loss on the current task $t$ , GEM treats the losses on the episodic memories of tasks $k \ < \ t$ , given by $\begin{array} { r l r } { \ell ( f _ { \theta } , { \mathcal M } _ { k } ) } & { = } & { \frac { 1 } { | { \mathcal M } _ { k } | } \sum _ { ( { \mathbf x } _ { i } , k , y _ { i } ) \in { \mathcal M } _ { k } } \ell ( f _ { \theta } ( { \mathbf x } _ { i } , k ) , y _ { i } ) } \end{array}$ , as inequality constraints, avoiding their increase but allowing their decrease. This effectively permits GEM to do positive backward transfer which other LLL methods do not support. Formally, at task $t$ , GEM solves for the following objective:
+
+$$
+\begin{array} { r l } { \mathrm { m i n i m i z e } _ { \theta } \quad \ell ( f _ { \theta } , \mathcal { D } _ { t } ) \quad \mathrm { s . t . } \quad \ell ( f _ { \theta } , \mathcal { M } _ { k } ) \leq \ell ( f _ { \theta } ^ { t - 1 } , \mathcal { M } _ { k } ) \qquad \forall k < t } \end{array}
+$$
+
+Where $f _ { \theta } ^ { t - 1 }$ is the network trained till task $t - 1$ . To inspect the increase in loss, GEM computes the angle between the loss gradient vectors of previous tasks $g _ { k }$ , and the proposed gradient update on the current task $g$ . Whenever the angle is greater than $9 0 ^ { \circ }$ with any of the $g _ { k }$ ’s, it projects the proposed gradient to the closest in L2 norm gradient $\tilde { g }$ that keeps the angle within the bounds. Formally, the optimization problem GEM solves is given by:
+
+$$
+\mathrm { m i n i m i z e } _ { \tilde { g } } \quad { \frac { 1 } { 2 } } | | g - \tilde { g } | | _ { 2 } ^ { 2 } \quad \mathrm { s . t . } \quad \langle \tilde { g } , g _ { k } \rangle \geq 0
+$$
+
+Eq.7 is a quadratic program (QP) in $P$ -variables (the number of parameters in the network), which for neural networks could be in millions. In order to solve this efficiently, GEM works in the dual space which results in a much smaller QP with only $t - 1$ variables:
+
+$$
+\mathrm { m i n i m i z e } _ { v } \quad { \frac { 1 } { 2 } } { v } ^ { \top } G G ^ { \top } v + g ^ { \top } G ^ { \top } v \quad \mathrm { s . t . } \quad v \geq 0
+$$
+
+where $G = - ( g _ { 1 } , \cdot \cdot \cdot , g _ { t - 1 } ) \in \mathbb { R } ^ { ( t - 1 ) \times P }$ is computed at each gradient step of training. Once the solution $v ^ { * }$ to Eq. 8 is found, the projected gradient update can be computed as $\tilde { g } = G ^ { \top } v ^ { * } + g$ .
+
+While GEM has proven very effective in a single epoch setting (Lopez-Paz & Ranzato, 2017), the performance gains come at a big computational burden at training time. At each training step, GEM computes the matrix $G$ using all samples from the episodic memory, and it also needs to solve the QP of Eq. 8. Unfortunately, this inner loop optimization becomes prohibitive when the size of $\mathcal { M }$ and the number of tasks is large, see Tab. 7 in Appendix for an empirical analysis. To alleviate the computational burden of GEM, next we propose a much more efficient version of GEM, called Averaged GEM (A-GEM).
+
+Whereas GEM ensures that at every training step the loss of each individual previous tasks, approximated by the samples in episodic memory, does not increase, A-GEM tries to ensure that at every training step the average episodic memory loss over the previous tasks does not increase. Formally, while learning task $t$ , the objective of A-GEM is:
+
+$$
+\operatorname { m i n i m i z e } _ { \boldsymbol { \theta } } \quad \ell ( f _ { \boldsymbol { \theta } } , \mathcal { D } _ { t } ) \quad \mathrm { s . t . } \quad \ell ( f _ { \boldsymbol { \theta } } , \mathcal { M } ) \leq \ell ( f _ { \boldsymbol { \theta } } ^ { t - 1 } , \mathcal { M } ) \qquad \mathrm { w h e r e ~ } \mathcal { M } = \cup _ { k < t } \mathcal { M } _ { k }
+$$
+
+The corresponding optimization problem reduces to:
+
+$$
+\mathrm { m i n i m i z e } _ { \tilde { g } } \quad \frac { 1 } { 2 } | | g - \tilde { g } | | _ { 2 } ^ { 2 } \quad \mathrm { s . t . } \quad \tilde { g } ^ { \top } g _ { r e f } \geq 0
+$$
+
+where $g _ { r e f }$ is a gradient computed using a batch randomly sampled from the episodic memory, $( \mathbf { x } _ { r e f } , y _ { r e f } ) \sim \mathcal { M }$ , of all the past tasks. In other words, A-GEM replaces the $t - 1$ constraints of GEM with a single constraint, where $g _ { r e f }$ is the average of the gradients from the previous tasks computed from a random subset of the episodic memory.
+
+The constrained optimization problem of Eq. 10 can now be solved very quickly; when the gradient $g$ violates the constraint, it is projected via:
+
+$$
+\tilde { g } = g - \frac { g ^ { \top } g _ { r e f } } { g _ { r e f } ^ { \top } g _ { r e f } } g _ { r e f }
+$$
+
+The formal proof of the update rule of A-GEM (Eq. 11) is given in Appendix C. This makes A-GEM not only memory efficient, as it does not need to store the matrix $G$ , but also orders of magnitude faster than GEM because 1) it is not required to compute the matrix $G$ but just the gradient of a random subset of memory examples, 2) it does not need to solve any QP but just an inner product, and 3) it will incur in less violations particularly when the number of tasks is large (see Tab. 7 and Fig. 6 in Appendix for empirical evidence). All together these factors make A-GEM faster while not hampering its good performance in the single pass setting.
+
+Intuitively, the difference between GEM and A-GEM loss functions is that GEM has better guarantess in terms of worst-case forgetting of each individual task since (at least on the memory examples) it prohibits an increase of any task-specific loss, while A-GEM has better guaratees in terms of average accuracy since GEM may prevent a gradient step because of a task constraint violation although the overall average loss may actually decrease, see Appendix Sec. D.1 and D.2 for further analysis and empirical evidence. The pseudo-code of A-GEM is given in Appendix Alg. 2.
+
+# 5 JOINT EMBEDDING MODEL USING COMPOSITIONAL TASK DESCRIPTORS
+
+In this section, we discuss how we can improve forward transfer for all the LLL methods including A-GEM. In order to speed up learning of a new task, we consider the use of compositional task descriptors where components are shared across tasks and thus allow transfer. Examples of compositional task descriptors are, for instance, a natural language description of the task under consideration or a matrix specifying the attribute values of the objects to be recognized in the task. In our experiments, we use the latter since it is provided with popular benchmark datasets (Wah et al., 2011; Lampert et al., 2009). For instance, if the model has already learned and remembers about two independent properties (e.g., color of feathers and shape of beak), it can quickly recognize a new class provided a descriptor specifying the values of its attributes (yellow feathers and red beak), although this is an entirely unseen combination.
+
+Borrowing ideas from literature in few-shot learning (Lampert et al., 2014; Zhang et al., 2018; Elhoseiny et al., 2017; Xian et al., 2018), we learn a joint embedding space between image features and the attribute embeddings. Formally, let $\mathbf { x } ^ { k } \in \breve { \mathcal { X } }$ be the input (e.g., an image), $t ^ { k }$ be the task descriptor in the form of a matrix of size $C _ { k } \times A$ , where $C _ { k }$ is the number of classes in the $k$ - th task and $A$ is the total number of attributes for each class in the dataset. The joint embedding model consists of a feature extraction module, $\phi _ { \theta } : \mathbf { x } ^ { k }  \phi _ { \theta } ( \mathbf { x } ^ { k } )$ , where $\phi _ { \theta } ( \mathbf { x } ^ { k } ) \doteq \mathbb { R } ^ { D }$ , and a task embedding module, $\psi _ { \omega } : t ^ { k } \to \psi _ { \omega } ( t ^ { k } )$ , where $\psi _ { \omega } ( t ^ { k } ) \in \mathbb { R } ^ { C _ { k } \times D }$ . In this work, $\phi _ { \theta } ( . )$ is implemented as a standard multi-layer feed-forward network (see Sec. 6 for the exact parameterization), whereas $\psi _ { \omega } ( . )$ is implemented as a parameter matrix of dimensions $A \times D$ . This matrix can be interpreted as an attribute look-up table as each attribute is associated with a $D$ dimensional vector, from which a class embedding vector is constructed via a linear combination of the attributes present in the class; the task descriptor embedding is then the concatenation of the embedding vectors of the classes present in the task (see Appendix Fig. 9 for the pictorial description of the joint embedding model). During training, the parameters $\theta$ and $\omega$ are learned by minimizing the cross-entropy loss:
+
+$$
+\ell _ { k } ( \theta , \omega ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } - \log ( p ( y _ { i } ^ { k } | \mathbf { x } _ { i } ^ { k } , t ^ { k } ; \theta , \omega ) )
+$$
+
+where $( \mathbf { x } _ { i } ^ { k } , t ^ { k } , y _ { i } ^ { k } )$ is the $i$ -th example of task $k$ . If $y _ { i } ^ { k } = c$ , then the distribution $p ( . )$ is given by:
+
+$$
+p ( c | \mathbf { x } _ { i } ^ { k } , t ^ { k } ; \theta , \omega ) = \frac { \exp ( [ \phi _ { \theta } ( \mathbf { x } _ { i } ^ { k } ) \psi _ { \omega } ( t ^ { k } ) ^ { \top } ] _ { c } ) } { \sum _ { j } \exp ( [ \phi _ { \theta } ( \mathbf { x } _ { i } ^ { k } ) \psi _ { \omega } ( t ^ { k } ) ^ { \top } ] _ { j } ) }
+$$
+
+where $[ a ] _ { i }$ denotes the $i$ -th element of the vector $a$ . Note that the architecture and loss functions are general, and apply not only to A-GEM but also to any other LLL model (e.g., regularization based approaches). See Sec. 6 for the actual choice of parameterization of these functions.
+
+# 6 EXPERIMENTS
+
+We consider four dataset streams, see Tab.1 in Appendix Sec. A for a summary of the statistics. Permuted MNIST (Kirkpatrick et al., 2016) is a variant of MNIST (LeCun, 1998) dataset of handwritten digits where each task has a certain random permutation of the input pixels which is applied to all the images of that task. Split CIFAR (Zenke et al., 2017) consists of splitting the original CIFAR-100 dataset (Krizhevsky & Hinton, 2009) into 20 disjoint subsets, where each subset is constructed by randomly sampling 5 classes without replacement from a total of 100 classes. Similarly to Split CIFAR, Split CUB is an incremental version of the fine-grained image classification dataset CUB (Wah et al., 2011) of 200 bird categories split into 20 disjoint subsets of classes. Split AWA, on the other hand, is the incremental version of the AWA dataset (Lampert et al., 2009) of 50 animal categories, where each task is constructed by sampling 5 classes with replacement from the total 50 classes, constructing 20 tasks. In this setting, classes may overlap among multiple tasks, but within each task they compete against different set of classes. Note that to make sure each training example is only seen once, the training data of a each class is split into disjoint sets depending on the frequency of its occurrence in different tasks. For Split AWA, the classifier weights of each class are randomly initialized within each head without any transfer from the previous occurrence of the class in past tasks. Finally, while on Permuted MNIST and Split CIFAR we provide integer task descriptors, on Split CUB and Split AWA we stack together the attributes of the classes (specifying for instance the type of beak, the color of feathers, etc.) belonging to the current task to form a descriptor.
+
+In terms of architectures, we use a fully-connected network with two hidden layers of 256 ReLU units each for Permuted MNIST, a reduced ResNet18 for Split CIFAR like in Lopez-Paz & Ranzato (2017), and a standard ResNet18 (He et al., 2016) for Split CUB and Split AWA. For a given dataset stream, all models use the same architecture, and all models are optimized via stochastic gradient descent with mini-batch size equal to 10. We refer to the joint-embedding model version of these models by appending the suffix ‘-JE’ to the method name.
+
+As described in Sec. 2 and outlined in Alg. 1, in order to cross validate we use the first 3 tasks, and then report metrics on the remaining 17 tasks after doing a single training pass over each task in sequence.
+
+Lastly, we compared A-GEM against several baselines and state-of-the-art LLL approaches which we describe next. VAN is a single supervised learning model, trained continually without any regularization, with the parameters of a new task initialized from the parameters of the previous task. ICARL (Rebuffi et al., 2017) is a class-incremental learner that uses nearest-exemplar-based classifier and avoids catastrophic forgetting by regularizing over the feature representation of previous tasks using a knowledge distillation loss. EWC (Kirkpatrick et al., 2016), PI (Zenke et al., 2017), RWALK (Chaudhry et al., 2018) and MAS (Aljundi et al., 2018) are regularization-based approaches aiming at avoiding catastrophic forgetting by limiting learning of parameters critical to the performance of past tasks. Progressive Networks (PROG-NN) (Rusu et al., 2016) is a modular approach whereby a new “column” with lateral connections to previous hidden layers is added once a new task arrives. GEM (Lopez-Paz & Ranzato, 2017) described in Sec. 4 is another natural baseline of comparison since A-GEM builds upon it. The amount of episodic memory per task used in ICARL, GEM and A-GEM is set to 250, 65, 50, and 100, and the batch size for the computation of $g _ { r e f }$ (when the episodic memory is sufficiently filled) in A-GEM is set to 256, 1300, 128 and 128 for MNIST, CIFAR, CUB and AWA, respectively. While populating episodic memory, the samples are chosen uniformly at random for each task. Whereas the network weights are randomly initialized for MNIST, CIFAR and AWA, on the other hand, for CUB, due to the small dataset size, a pre-trained ImageNet model is used. Finally, we consider a multi-task baseline, MULTI-TASK, trained on a single pass over shuffled data from all tasks, and thus violating the LLL assumption. It can be seen as an upper bound performance for average accuracy.
+
+![](images/a31e6bf25bbad1939f9629625b6552c2c4dbdd7c2ef233a6d338f91dad450e7d.jpg)  
+Figure 1: Performance of LLL models across different measures on Permuted MNIST and Split CIFAR. For Accuracy $( A _ { T } )$ and Learning Curve Measure $( L C A _ { 1 0 } )$ the higher the number (indicated by ↑) the better is the model. For Forgetting $( F _ { T } )$ , Time and Memory the lower the number (indicated by ↓) the better is the model. For Time and Memory, the method with the highest complexity is taken as a reference (value of 1) and the other methods are reported relative to that method. $A _ { T }$ , $F _ { T }$ and $L C A _ { 1 0 }$ values and confidence intervals are computed over 5 runs. A-GEM provides the best trade-off across different measures and dimensions. Other baselines are given in Tab. 4 and 7 in the Appendix, which are used to generate the plots.
+
+![](images/0f15052f26c48ea5e8fb8a44697683e3f4cbc6cac5cbe875b9ac04917901f715.jpg)  
+Figure 2: Performance of LLL models across different measures on Split CUB and Split AWA. On both the datasets, PROG-NN runs out of memory. The memory and time complexities of joint embedding models are the same as those of the corresponding standard models and are hence omitted. $A _ { T }$ , $F _ { T }$ and $L C A _ { 1 0 }$ values and confidence intervals are computed over 10 runs. Other baselines are given in Tab. 5, 6 and 7 in the Appendix, which are used to generate the plots.
+
+# 6.1 RESULTS
+
+Fig. 1 and 2 show the overall results on all the datasets we considered (for brevity we show only representative methods, see detailed results in Appendix Tab. 4, 5, 6 and 7). First, we observe that A-GEM achieves the best average accuracy on all datasets, except Permuted MNIST, where PROGNN works better. The reason is because on this dataset each task has a large number of training examples, which enables PROG-NN to learn its task specific parameters and to leverage its lateral connections. However, notice how PROG-NN has the worst memory cost by the end of training - as its number of parameters grows super-linearly with the number of tasks. In particular, in large scale setups (Split CUB and AWA), PROG-NN runs out of memory during training due to its large size. Also, PROG-NN does not learn well on datasets where tasks have fewer training examples. Second, A-GEM and GEM perform comparably in terms of average accuracy, but A-GEM has much lower time (about 100 times faster) and memory cost (about 10 times lower), comparable to regularizationbased approaches like EWC. Third, EWC and similar methods perform only slightly better than VAN on this single pass LLL setting. The analysis in Appendix Sec. F demonstrates that EWC requires several epochs and over-parameterized architectures in order to work well. Fourth, PROG-NN has no forgetting by construction and A-GEM and GEM have the lowest forgetting among methods that use a fixed capacity architecture. Next, all methods perform similarly in terms of LCA, with PROGNN being the worst because of its ever growing number of parameters and A-GEM slightly better than all the other approaches. And finally, the use of task descriptors improves average accuracy across the board as shown in Fig.2, with A-GEM a bit better than all the other methods we tried. All joint-embedding models using task descriptors have better LCA performance, although this is the same across all methods including A-GEM. Overall, we conclude that A-GEM offers the best trade-off between average accuracy performance and efficiency in terms of sample, memory and computational cost.
+
+![](images/0f85b86c3a7876b61b6b28dafe04a4b9ee8e26dbba4d681da496e91770e9fb61.jpg)  
+Figure 3: Top Row: Evolution of average accuracy $( A _ { k } )$ as new tasks are learned. Bottom Row: Evolution of LCA during the first ten mini-batches.
+
+![](images/04b36ea073776c12611d463920ed118425e1c1e24720ca913773729b6ef66f68.jpg)  
+Figure 4: Evolution of LCA during the first ten mini-batches.
+
+Fig. 3 shows a more fine-grained analysis and comparison with more methods on Permuted MNIST and Split CIFAR. The average accuracy plots show how A-GEM and GEM greatly outperform other approaches, with the exception of PROG-NN on MNIST as discussed above. On different datasets, different methods are best in terms of LCA, although A-GEM is always top-performing. Fig. 4 shows in more detail the gain brought by task descriptors which greatly speed up learning in the few-shot regime. On these datasets, A-GEM performs the best or on par to the best.
+
+![](images/94132bf2931f255a034e2078abe892843bd98492500194e4837678d985acf63d.jpg)  
+Figure 5: Evolution of zero-shot performance as the learner sees new tasks on Split CUB and Split AWA datasets.
+
+Finally, in Fig. 5, we report the 0-shot performance of LLL methods on Split CUB and Split AWA datasets over time, showing a clear advantage of using compositional task descriptors with joint embedding models, which is more significant for A-GEM. Interestingly, the zero-shot learning performance of joint embedding models improves over time, indicating that these models get better at forward transfer or, in other words, become more efficient over time.
+
+# 7 RELATED WORK
+
+Continual (Ring, 1997) or Lifelong Learning (LLL) (Thrun, 1998) have been the subject of extensive study over the past two decades. One approach to LLL uses modular compositional models (Fernando et al., 2017; Aljundi et al., 2017; Rosenbaum et al., 2018; Chang et al., 2018; Xu & Zhu, 2018; Ferran Alet, 2018), which limit interference among tasks by using different subset of modules for each task. Unfortunately, these methods require searching over the space of architectures which is not sample efficient with current methods. Another approach is to regularize parameters important to solve past tasks (Kirkpatrick et al., 2016; Zenke et al., 2017; Chaudhry et al., 2018), which has been proven effective for over-parameterized models in the multiple epoch setting (see Appendix Sec. F), while we focus on learning from few examples using memory efficient models. Methods based on episodic memory (Rebuffi et al., 2017; Lopez-Paz & Ranzato, 2017) require a little bit more memory at training time but can work much better in the single pass setting we considered (Lopez-Paz & Ranzato, 2017).
+
+The use of task descriptors for LLL has already been advocated by Isele et al. (2016) but using a sparse coding framework which is not obviously applicable to deep nets in a computationally efficient way, and also by Lopez-Paz & Ranzato (2017) although they did not explore the use of compositional descriptors. More generally, tasks descriptors have been used in Reinforcement Learning with similar motivations by several others (Sutton et al., 2011; Schaul et al., 2015; Baroni et al., 2017), and it is also a key ingredient in all the zero/few-shot learning algorithms (Lampert et al., 2014; Xian et al., 2018; Elhoseiny et al., 2017; Wah et al., 2011; Lampert et al., 2009).
+
+# 8 CONCLUSION
+
+We studied the problem of efficient Lifelong Learning (LLL) in the case where the learner can only do a single pass over the input data stream. We found that our approach, A-GEM, has the best tradeoff between average accuracy by the end of the learning experience and computational/memory cost. Compared to the original GEM algorithm, A-GEM is about 100 times faster and has 10 times less memory requirements; compared to regularization based approaches, it achieves significantly higher average accuracy. We also demonstrated that by using compositional task descriptors all methods can improve their few-shot performance, with A-GEM often being the best.
+
+Our detailed experiments reported in Appendix E also show that there is still a substantial performance gap between LLL methods, including A-GEM, trained in a sequential learning setting and the same network trained in a non-sequential multi-task setting, despite seeing the same data samples. Moreover, while task descriptors do help in the few-shot learning regime, the LCA performance gap between different methods is very small; suggesting a poor ability of current methods to transfer knowledge even when forgetting has been eliminated. Addressing these two fundamental issues will be the focus of our future research.
+
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+
+# APPENDIX
+
+In Sec. A we report the summary of datasets used for the experiments. Sec. B details our A-GEM algorithm and Sec. C provides the proof of update rule of A-GEM discussed in Sec. 4 of the main paper. In Sec. D, we analyze the differences between A-GEM and GEM, and describe another variation of GEM, dubbed Stochastic GEM (S-GEM). The detailed results of the experiments which were used to generate Fig 1 and 2 in the main paper are given in Sec. E. In Sec. F, we provide empirical evidence to the conjecture that regularization-based approaches like EWC require over-parameterized architectures and multiple passes over data in order to perform well as discussed in the Sec. 6.1 of the main paper. In Sec. G, we provide the grid used for the cross-validation of different hyperparameters and report the optimal values for different models. Finally, in Sec. H, we pictorially describe the joint embedding model discussed in Sec. 5.
+
+# A DATASET STATISTICS
+
+Table 1: Dataset statistics.   
+
+<table><tr><td></td><td>Perm.MNIST</td><td>Split CIFAR</td><td>Split CUB</td><td>Split AWA</td></tr><tr><td>num.of tasks</td><td>20</td><td>20</td><td>20</td><td>20</td></tr><tr><td>input size</td><td>1×28×28</td><td>3×32×32</td><td>3×224×224</td><td>3×224×224</td></tr><tr><td>num. of classes per task</td><td>10</td><td>5</td><td>10</td><td>5</td></tr><tr><td>num.of training images per task</td><td>60000</td><td>2500</td><td>300</td><td>1</td></tr><tr><td>num.of test images per task</td><td>10000</td><td>500</td><td>290</td><td>560</td></tr></table>
+
+# B A-GEM ALGORITHM
+
+# Algorithm 2 Training and evaluation of A-GEM on sequential data $\mathcal { D } = \{ \mathcal { D } _ { 1 } , \cdot \cdot \cdot , \mathcal { D } _ { T } \}$
+
+1: procedure TRAIN(fθ, Dtrain, Dtest)   
+2: M ← {}   
+3: A ← 0 ∈ RT ×T   
+4: for $t = \{ 1 , \cdots , T \}$ do   
+5: for $( \mathbf { x } , y ) \in \mathcal { D } _ { t } ^ { t r a i n }$ do   
+6: $( \mathbf { x } _ { r e f } , y _ { r e f } ) \sim \mathcal { M }$   
+7: $g _ { r e f } \gets \nabla _ { \theta } \ell ( f _ { \theta } ( \mathbf { x } _ { r e f } , t ) , y _ { r e f } )$   
+8: g ← ∇θ\`(fθ(x, t), y)   
+9: if g>gref ≥ 0 then   
+10: g˜ ← g   
+11: else   
+12: 13: g˜ ← g $\begin{array} { r } { \tilde { g } \gets g - \frac { g ^ { \top } g _ { r e f } } { g _ { r e f } ^ { \top } g _ { r e f } } } { g _ { r e f } g _ { r e f } }  \end{array}$   
+14: θ ← θ − αg˜   
+15: end for   
+16: M ← UPDATEEPSMEM(M, Dtraint , T )   
+17: At,: ← EVAL(fθ, Dtest)   
+18: end for   
+19: return $f _ { \theta , A }$   
+20: end procedure   
+1: procedure EVAL(fθ, Dtest)   
+2: a ← 0 ∈ RT   
+3: for t = {1, · · · , T } do   
+4: at ← 0   
+5: for (x, y) ∈ Dtestt do   
+6: at ← at + ACCURACY(fθ(x, t), y)   
+7: end for   
+8: $\begin{array} { r } { a _ { t } \gets \frac { a _ { t } } { l e n ( \mathcal { D } _ { t } ^ { t e s t } ) } } \end{array}$   
+9: end for   
+10: return $a$   
+11: end procedure   
+1: procedure UPDATEEPSMEM $\mathsf { \Omega } _ { [ } ( \mathcal { M } , \mathcal { D } _ { t } , T )$   
+2: $s  { \frac { \vert { \mathcal { M } } \vert } { T } }$   
+3: for i = {1, · · · , s} do   
+4: (x, y) ∼ Dt   
+5: M ← (x, y)   
+6: end for   
+7: return $\mathcal { M }$   
+8: end procedure
+
+# C A-GEM UPDATE RULE
+
+Here we provide the proof of the update rule of A-GEM (Eq. 11), $\begin{array} { r } { \tilde { g } = g - \frac { g ^ { \top } g _ { r e f } } { g _ { r e f } ^ { \top } g _ { r e f } } g _ { r e f } } \end{array}$ − g grefg> gref gref , stated in Sec. 4 of the main paper.
+
+Proof. The optimization objective of A-GEM as described in the Eq. 10 of the main paper, is:
+
+$$
+\begin{array} { r l } { \operatorname* { m i n i m i z e } _ { \tilde { g } } } & { \frac { 1 } { 2 } | | g - \tilde { g } | | _ { 2 } ^ { 2 } } \\ { \mathrm { s . t . } } & { \tilde { g } ^ { \top } g _ { r e f } \geq 0 } \end{array}
+$$
+
+Replacing $\tilde { g }$ with $z$ and rewriting Eq. 14 yields:
+
+$$
+\begin{array} { r l } { \operatorname * { m i n i m i z e } _ { z } } & { { } \frac { 1 } { 2 } z ^ { \top } z - g ^ { \top } z } \\ { \mathrm { s . t . } } & { { } - z ^ { \top } g _ { r e f } \leq 0 } \end{array}
+$$
+
+Note that we discard the term $g ^ { \intercal } g$ from the objective and change the sign of the inequality constraint. The Lagrangian of the constrained optimization problem defined above can be written as:
+
+$$
+\mathcal { L } ( z , \alpha ) = \frac { 1 } { 2 } z ^ { \top } z - g ^ { \top } z - \alpha z ^ { \top } g _ { r e f }
+$$
+
+Now, we pose the dual of Eq. 16 as:
+
+$$
+\theta _ { \mathcal D } ( \alpha ) = \operatorname* { m i n } _ { z } \mathcal L ( z , \alpha )
+$$
+
+Lets find the value $z ^ { * }$ that minimizes the $\mathcal { L } ( z , \alpha )$ by setting the derivatives of $\mathcal { L } ( z , \alpha )$ w.r.t. to $z$ to zero:
+
+$$
+\begin{array} { r } { \nabla _ { z } \mathcal { L } ( z , \alpha ) = 0 } \\ { z ^ { * } = g + \alpha g _ { r e f } } \end{array}
+$$
+
+The simplified dual after putting the value of $z ^ { * }$ in Eq. 17 can be written as:
+
+$$
+\begin{array} { l } { { \displaystyle \theta _ { D } ( \alpha ) = \frac { 1 } { 2 } ( g ^ { \top } g + 2 \alpha g ^ { \top } g _ { r e f } + \alpha ^ { 2 } g _ { r e f } ^ { \top } g _ { r e f } ) - g ^ { \top } g - 2 \alpha g ^ { \top } g _ { r e f } - \alpha ^ { 2 } g _ { r e f } ^ { \top } g _ { r e f } } } \\ { { \displaystyle \quad \quad = - \frac { 1 } { 2 } g ^ { \top } g - \alpha g ^ { \top } g _ { r e f } - \frac { 1 } { 2 } \alpha ^ { 2 } g _ { r e f } ^ { \top } g _ { r e f } } } \end{array}
+$$
+
+The solution $\begin{array} { r } { \alpha ^ { * } = \operatorname* { m a x } _ { \alpha ; \alpha > 0 } \theta _ { \mathcal { D } } ( \alpha ) } \end{array}$ to the dual is given by:
+
+$$
+\begin{array} { r } { \nabla _ { \alpha } \theta _ { \mathcal { D } } ( \alpha ) = 0 } \\ { \alpha ^ { * } = - \frac { g ^ { \top } g _ { r e f } } { g _ { r e f } ^ { \top } g _ { r e f } } } \end{array}
+$$
+
+By putting $\alpha ^ { * }$ in Eq. 18, we recover the A-GEM update rule:
+
+$$
+z ^ { * } = g - \frac { g ^ { \top } g _ { r e f } } { g _ { r e f } ^ { \top } g _ { r e f } } g _ { r e f } = \tilde { g }
+$$
+
+# D ANALYSIS OF GEM AND A-GEM
+
+In this section, we empirically analyze the differences between A-GEM and GEM, and report experiments with another computationally efficient but worse performing version of GEM.
+
+# D.1 FREQUENCY OF CONSTRAINT VIOLATIONS
+
+Fig. 6 shows the frequency of constraint violations (see Eq. 8 and 10) on Permuted MNIST and Split CIFAR datasets. Note that, the number of gradient updates (training steps) per task on MNIST and CIFAR are 5500 and 250, respectively. As the number of tasks increase, GEM violates the optimization constraints at almost each training step, whereas A-GEM plateaus to a much lower value. Therefore, the computational efficiency of A-GEM not only stems from the fact that it avoids solving a QP at each training step (which is much more expensive than a simple inner product) but also from the fewer number of constraint violations. From the figure, we can also infer that as the number of tasks grows the gap between GEM and A-GEM would grow further. Thus, the computational and memory overhead of GEM over A-GEM, see also Tab. 7, gets worse as the number of tasks increases.
+
+![](images/11de576c0a305ef6ed772507c504abd22d6213a6852c5a81bd046c65745080d4.jpg)  
+Figure 6: Number of constraint violations in GEM and A-GEM on Permuted MNIST and Split CIFAR as new tasks are learned.
+
+D.2 AVERAGE ACCURACY AND WORST-CASE FORGETTING
+
+In Tab. 2, we empirically demonstrate the different properties induced by the objective functions of GEM and A-GEM. GEM enjoys lower worst-case task forgetting while A-GEM enjoys better overall average accuracy. This is particularly true on the training examples stored in memory, as on the test set the result is confounded by the generalization error.
+
+Table 2: Comparison of average accuracy $( A _ { T } )$ and worst-case forgetting $( F _ { w s t } )$ on the Episodic Memory $( \mathcal { M } )$ and Test Set $( \mathcal { D } ^ { \check { E } V } )$ .   
+
+<table><tr><td>Methods</td><td colspan="4">MNIST</td><td colspan="4">CIFAR</td></tr><tr><td rowspan="2"></td><td colspan="2">M</td><td colspan="2">DEV</td><td colspan="2">M</td><td colspan="2">DEV</td></tr><tr><td>AT</td><td>Fwst</td><td>AT</td><td>Fwst</td><td>Ar</td><td>Fwst</td><td>AT</td><td>Fwst</td></tr><tr><td>GEM</td><td>99.5</td><td>0</td><td>89.5</td><td>0.10</td><td>97.1</td><td>0.05</td><td>61.2</td><td>0.14</td></tr><tr><td>A-GEM</td><td>99.3</td><td>0.008</td><td>89.1</td><td>0.13</td><td>72.1</td><td>0.15</td><td>62.3</td><td>0.15</td></tr></table>
+
+# D.3 STOCHASTIC GEM (S-GEM)
+
+In this section we report experiments with another variant of GEM, dubbed Stochastic GEM (SGEM). The main idea in S-GEM is to randomly sample one constraint, at each training step, from the possible $t - 1$ constraints of GEM. If that constraint is violated, the gradient is projected only taking into account that constraint. Formally, the optimization objective of S-GEM is given by:
+
+$$
+\begin{array} { r l } { \operatorname * { m i n i m i z e } _ { \tilde { g } } } & { \displaystyle \frac { 1 } { 2 } | | g - \tilde { g } | | _ { 2 } ^ { 2 } } \\ { \mathrm { s . t . } } & { \langle \tilde { g } , g _ { k } \rangle \geq 0 \quad \mathrm { w h e r e } \ k \sim \{ 1 , \cdots , t - 1 \} } \end{array}
+$$
+
+In other words, at each training step, S-GEM avoids the increase in loss of one of the previous tasks sampled randomly. In Tab. 3 we report the comparison of GEM, S-GEM and A-GEM on Permuted MNIST and Split CIFAR.
+
+Although, S-GEM is closer in spirit to GEM, as it requires randomly sampling one of the GEM constraints to satisfy, compared to A-GEM, which defines the constraint as the average gradient of the previous tasks, it perform slightly worse than GEM, as can be seen from Tab. 3.
+
+Table 3: Comparison of different variations of GEM on MNIST Permutations and Split CIFAR.   
+
+<table><tr><td>Methods</td><td colspan="2">Permuted MNIST</td><td colspan="2">Split CIFAR</td></tr><tr><td></td><td>Ar(%)</td><td>FT</td><td>Ar(%)</td><td>FT</td></tr><tr><td>GEM</td><td>89.5</td><td>0.06</td><td>61.2</td><td>0.06</td></tr><tr><td>S-GEM</td><td>88.2</td><td>0.08</td><td>56.2</td><td>0.12</td></tr><tr><td>A-GEM</td><td>89.1</td><td>0.06</td><td>62.3</td><td>0.07</td></tr></table>
+
+# E RESULT TABLES
+
+In Tab. 4, 5, 6 and 7 we report the detailed results which were used to generate Fig.1 and 2.
+
+Table 4: Comparison with different baselines on Permuted MNIST and Split CIFAR. The value of $\infty$ is assigned to a metric when the model fails to train with the cross-validated values of hyperparameters found on the subset of the tasks as discussed in Sec. 2 of the main paper. The numbers are averaged across 5 runs using a different seed each time. The results from this table are used to generate Fig 1 in Sec. 6.1 of the main paper.   
+
+<table><tr><td>Methods</td><td colspan="3">Permuted MNIST</td><td colspan="3">Split CIFAR</td></tr><tr><td></td><td>Ar(%)</td><td>FT</td><td>LCA10</td><td>Ar(%)</td><td>FT</td><td>LCA10</td></tr><tr><td>VAN</td><td>47.9 (± 1.32)</td><td>0.51 (± 0.01)</td><td>0.26 (± 0.006)</td><td>42.9 (± 2.07)</td><td>0.25 (± 0.03)</td><td>0.30 (± 0.008)</td></tr><tr><td>ICARL</td><td></td><td></td><td></td><td>50.1</td><td>0.11</td><td></td></tr><tr><td>EWC</td><td>68.3 (± 0.69)</td><td>0.29 (± 0.01)</td><td>0.27 (± 0.003)</td><td>42.4 (± 3.02)</td><td>0.26 (± 0.02)</td><td>0.33 (± 0.01)</td></tr><tr><td>PI</td><td>8</td><td>8</td><td>8</td><td>47.1 (± 4.41)</td><td>0.17 (± 0.04)</td><td>0.31 (± 0.008)</td></tr><tr><td>MAS</td><td>69.6 (± 0.93)</td><td>0.27 (± 0.01)</td><td>0.29 (± 0.003)</td><td>44.2 (± 2.39)</td><td>0.25 (± 0.02)</td><td>0.33 (± 0.009)</td></tr><tr><td>RWALK</td><td>85.7 (± 0.56)</td><td>0.08 (± 0.01)</td><td>0.31 (± 0.005)</td><td>40.9 (± 3.97)</td><td>0.29 (± 0.04)</td><td>0.32 (± 0.005)</td></tr><tr><td>PROG-NN</td><td>93.5 (± 0.07)</td><td>0</td><td>0.19 (± 0.006)</td><td>59.2 (± 0.85)</td><td>0</td><td>0.21 (± 0.001)</td></tr><tr><td>GEM</td><td>89.5 (± 0.48)</td><td>0.06 (± 0.004)</td><td>0.23 (± 0.005)</td><td>61.2 (± 0.78)</td><td>0.06 (± 0.007)</td><td>0.36 (± 0.007)</td></tr><tr><td>A-GEM (Ours)</td><td>89.1 (± 0.14)</td><td>0.06 (± 0.001)</td><td>0.29 (± 0.004)</td><td>62.3 (± 1.24)</td><td>0.07 (± 0.01)</td><td>0.35 (± 0.01)</td></tr><tr><td>MULTI-TASK</td><td>95.3</td><td>-</td><td>=</td><td>68.3</td><td>-</td><td>-</td></tr></table>
+
+Table 5: Average accuracy and forgetting of standard models (left) and joint embedding models (right) on Split CUB. The value of ‘OoM’ is assigned to a metric when the model fails to fit in the memory. The numbers are averaged across 10 runs using a different seed each time. The results from this table are used to generate Fig 2 in Sec. 6.1 of the main paper.   
+
+<table><tr><td>Methods</td><td colspan="3">Split CUB</td></tr><tr><td></td><td>Ar(%)</td><td>FT</td><td>LCA10</td></tr><tr><td>VAN</td><td>54.3 (± 2.03)/67.1 (± 4.77)</td><td>0.13 (± 0.02)/ 0.10 (± 0.04)</td><td>0.29 (± 0.009)/0.52 (± 0.01)</td></tr><tr><td>EWC</td><td>54 (±1.08)/68.4(± 4.08)</td><td>0.13 (± 0.02)/0.09 (± 0.03)</td><td>0.29 (± 0.007)/0.52 (± 0.01)</td></tr><tr><td>PI</td><td>55.3 (± 2.28) / 66.6 (± 5.18)</td><td>0.12 (±0.02)/0.10 (±0.04)</td><td>0.29 (± 0.008)/0.52 (± 0.01)</td></tr><tr><td>RWALK</td><td>54.4 (± 1.82) /67.4 (± 3.50)</td><td>0.13 (± 0.01)/0.10 (± 0.03)</td><td>0.29 (± 0.008)/0.52 (± 0.01)</td></tr><tr><td>PROG-NN</td><td>OoM/OoM</td><td>OoM/OoM</td><td>OoM/OoM</td></tr><tr><td>A-GEM (Ours)</td><td>62 (± 3.5) /71(± 2.83)</td><td>0.07 (± 0.02)/0.07 (± 0.01)</td><td>0.30 (± 0.008)/0.54 (± 0.02)</td></tr><tr><td>MULTI-TASK</td><td>65.6 / 73.8</td><td>-/-</td><td>-/-</td></tr></table>
+
+Table 6: Average accuracy and forgetting of standard models (left) and joint embedding models (right) on Split AWA. The value of ‘OoM’ is assigned to a metric when the model fails to fit in the memory. The numbers are averaged across 10 runs using a different seed each time. The results from this table are used to generate Fig 2 in Sec. 6.1 of the main paper.   
+
+<table><tr><td>Methods</td><td colspan="3">Split AWA</td></tr><tr><td></td><td>Ar(%)</td><td>FT</td><td>LCA10</td></tr><tr><td>VAN</td><td>30.3 (± 2.84)/42.8 (± 2.86)</td><td>0.04 (± 0.01)/0.07 (± 0.02)</td><td>0.21 (± 0.008)/0.37 (±0.02)</td></tr><tr><td>EWC</td><td>33.9 (± 2.87) /43.3 (± 3.71)</td><td>0.08 (± 0.02)/0.07 (± 0.03)</td><td>0.26 (± 0.01)/ 0.37 (± 0.02)</td></tr><tr><td>PI</td><td>33.9 (± 3.25)/43.4 (±3.49)</td><td>0.08 (± 0.02)/0.06 (±0.02)</td><td>0.26 (± 0.01)/0.37 (±0.02)</td></tr><tr><td>RWALK</td><td>33.9 (± 2.91)/42.9 (± 3.10)</td><td>0.08 (± 0.02)/0.07 (±0.02)</td><td>0.26 (± 0.01) / 0.37 (± 0.02)</td></tr><tr><td>PROG-NN</td><td>OoM/OoM</td><td>OoM/OoM</td><td>OoM/OoM</td></tr><tr><td>A-GEM (Ours)</td><td>44 (± 4.10) / 50 (± 3.25)</td><td>0.05 (± 0.02) /0.03 (± 0.02)</td><td>0.29 (± 0.01) / 0.39 (± 0.02)</td></tr><tr><td>MULTI-TASK</td><td>64.8 / 66.8</td><td>-/-</td><td>-/-</td></tr></table>
+
+Table 7: Computational cost and memory complexity of different LLL approaches. The timing refers to training time on a GPU device. Memory cost is provided in terms of the total number of parameters P, the size of the minibatch B, the total size of the network hidden state $H$ (assuming all methods use the same architecture), the size of the episodic memory M per task. The results from this table are used to generate Fig. 1 and 2 in Sec. 6.1 of the main paper.   
+
+<table><tr><td>Methods</td><td colspan="4">Training Time [s]</td><td colspan="2">Memory</td></tr><tr><td></td><td>MNIST</td><td>CIFAR</td><td>CUB</td><td>AWA</td><td>Training</td><td>Testing</td></tr><tr><td>VAN</td><td>186</td><td>105</td><td>54</td><td>4123</td><td>P +B*H</td><td>P+B*H</td></tr><tr><td>EWC</td><td>403</td><td>250</td><td>72</td><td>4136</td><td>4*P+B*H</td><td>P +B*H</td></tr><tr><td>PROGRESSIVE NETS</td><td>510</td><td>409</td><td>8</td><td>8</td><td>2*P*T + B*H*T</td><td>2*P*T+B*H*T</td></tr><tr><td>GEM</td><td>3442</td><td>5238</td><td>-</td><td>-</td><td>P*T + (B+M)*H</td><td>P +B*H</td></tr><tr><td>A-GEM (Ours)</td><td>477</td><td>449</td><td>420</td><td>5221</td><td>2*P + (B+M)*H</td><td>P +B*H</td></tr></table>
+
+# F ANALYSIS OF EWC
+
+In this section we provide empirical evidence to the conjecture that regularization-based approaches like EWC need over-parameterized architectures and multiple passes over the samples of each task in order to perform well. The intuition as to why models need to be over-parameterized is because it is easier to avoid cross-task interference when the model has additional capacity. In the single-pass setting and when each task does not have very many training samples, regularization-based appraches also suffer because regularization parameters cannot be estimated well from a model that has not fully converged. Moreover, for tasks that do not have much data, rgularization-based approaches do not enable any kind of positive backward transfer (Lopez-Paz & Ranzato, 2017) which further hurts performance as the predictor cannot leverage knowledge acquired later to improve its prediction on past tasks. Finally, regularization-based approaches perform much better in the multi-epoch setting simply because in this setting the baseline un-regularized model performs much worse, as it overfits much more to the data of the current task, every time unlearning what it learned before.
+
+We consider Permuted MNIST and Split CIFAR datasets as described in Sec. 6 of the main paper. For MNIST, the two architecture variants that we experiment with are; 1) two-layer fully-connected network with 256 units in each layer (denoted by $- S$ suffix), and 2) two-layer fully-connected network with 2000 units in each layer (denoted by $- B$ suffix).
+
+For CIFAR, the two architecture variants are; 1) ResNet-18 with 3 times less feature maps in all the layers (denoted by $- S$ suffix), and 2) Standard ResNet-18 (denoted by $- B$ token).
+
+We run the experiments on VAN and EWC with increasing the number of epochs from 1 to 10 for Permuted MNIST and from 1 to 30 for CIFAR. For instance, when epoch is set to 10, it means that the training samples of task $t$ are presented 10 times before showing examples from task $t + 1$ . In Fig. 7 and 8 we plot the Average Accuracy (Eq. 1) and Forgetting (Eq. 2) on Permuted MNIST and Split CIFAR, respectively.
+
+We observe that the average accuracy significantly improves with the number of epochs only when EWC is applied to the big network. In particular, in the single epoch setting, EWC peforms similarly to the baseline VAN on Split CIFAR which has fewer number of training examples per task.
+
+![](images/827122b242f9c02719c1d8c4cbc9f2ab63d03d83556a22db119b433dbc3e22d8.jpg)  
+Figure 7: Permuted MNIST: Change in average accuracy and forgetting as the number of epochs are increased. Tokens $\because S ^ { \prime }$ and $\mathbf { \epsilon } _ { - B } ,$ denote smaller and bigger networks, respectively.
+
+![](images/98fdcbd533faa39646c4b2f6d1aaa2c22a3fdddf0adeb2950c58c5ea70b94885.jpg)  
+Figure 8: Split CIFAR: Change in average accuracy and forgetting as the number of epochs are increased. Tokens $\because S ^ { \prime }$ and $\mathbf { \epsilon } _ { - B } ,$ denote smaller and bigger networks, respectively.
+
+# G HYPER-PARAMETER SELECTION
+
+Below we report the hyper-parameters grid considered for different experiments. Note, as described in the Sec. 6 of the main paper, to satisfy the requirement that a learner does not see the data of a task more than once, first $T ^ { C V }$ tasks are used to cross-validate the hyper-parameters. In all the datasets, the value of $T ^ { C V }$ is set to ‘3’. The best setting for each experiment is reported in the parenthesis.
+
+• MULTI-TASK – learning rate: [0.3, 0.1, 0.03 (MNIST perm, Split CIFAR, Split CUB, Split AWA), 0.01, 0.003, 0.001, 0.0003, 0.0001]   
+• MULTI-TASK-JE – learning rate: [0.3, 0.1, 0.03 (Split CUB, Split AWA), 0.01, 0.003, 0.001, 0.0003, 0.0001]   
+• VAN – learning rate: [0.3, 0.1, 0.03 (MNIST perm, Split CUB), 0.01 (Split CIFAR), 0.003, 0.001 (Split AWA), 0.0003, 0.0001]   
+• VAN-JE – learning rate: [0.3, 0.1, 0.03 (Split CUB), 0.01, 0.003 (Split AWA), 0.001, 0.0003, 0.0001]   
+• PROG-NN – learning rate: [0.3, 0.1 (MNIST perm, ), 0.03 (Split CIFAR, Split AWA), 0.01 (Split CUB), 0.003, 0.001, 0.0003, 0.0001]   
+• EWC – learning rate: [0.3, 0.1, 0.03 (MNIST perm, Split CIFAR, Split CUB), 0.01, 0.003 (Split AWA), 0.001, 0.0003, 0.0001]
+
+– regularization: [1 (Split CUB), 10 (MNIST perm, Split CIFAR), 100 (Split AWA), 1000, 10000]
+
+• EWC-JE – learning rate: [0.3, 0.1, 0.03 (Split CUB), 0.01, 0.003 (Split AWA), 0.001, 0.0003, 0.0001] – regularization: [1, 10 (Split CUB), 100 (Split AWA), 1000, 10000]
+
+P I – learning rate: [0.3, 0.1 (MNIST perm), 0.03 (Split CUB), 0.01 (Split CIFAR), 0.003 (Split AWA), 0.001, 0.0003, 0.0001] – regularization: [0.001, 0.01, 0.1 (MNIST perm, Split CIFAR, Split CUB), 1 (Split AWA), 10]
+
+• PI-JE
+
+– learning rate: [0.3, 0.1, 0.03 (Split CUB), 0.01, 0.003 (Split AWA), 0.001, 0.0003, 0.0001] – regularization: [0.001, 0.01, 0.1 (Split CUB), 1, 10 (Split AWA)]
+
+• MAS – learning rate: [0.3, 0.1 (MNIST perm), 0.03 (Split CIFAR, Split CUB), 0.01, 0.003 (Split AWA), 0.001, 0.0003, 0.0001] – regularization: [0.01, 0.1 (MNIST perm, Split CIFAR, Split CUB), 1 (Split AWA), 10]
+
+• MAS-JE
+
+– learning rate: [0.3, 0.1, 0.03 (Split CUB), 0.01, 0.003, 0.001 (Split AWA), 0.0003, 0.0001]   
+– regularization: [0.01, 0.1 (Split CUB, Split AWA), 1, 10]
+
+• RWALK
+
+– learning rate: [0.3, 0.1 (MNIST perm), 0.03 (Split CIFAR, Split CUB), 0.01, 0.003 (Split AWA), 0.001, 0.0003, 0.0001]   
+– regularization: [0.1, 1 (MNIST perm, Split CIFAR, Split CUB), 10 (Split AWA), 100, 1000]
+
+• RWALK-JE
+
+– learning rate: [0.3, 0.1, 0.03 (SPLIT CUB), 0.01, 0.003 (Split AWA), 0.001, 0.0003, 0.0001] – regularization: [0.1, 1 (Split CUB), 10 (Split AWA), 100, 1000]
+
+• A-GEM – learning rate: [0.3, 0.1 (MNIST perm), 0.03 (Split CIFAR, Split CUB), 0.01 (Split AWA), 0.003, 0.001, 0.0003, 0.0001]
+
+• A-GEM-JE – learning rate: [0.3, 0.1, 0.03 (SPLIT CUB), 0.01, 0.003 (Split AWA), 0.001, 0.0003, 0.0001]
+
+# H PICTORIAL DESCRIPTION OF JOINT EMBEDDING MODEL
+
+In Fig. 9 we provide a pictorial description of the joint embedding model discussed in the Sec. 5 of the main paper.
+
+![](images/eb9814ca8c064b11916d7438e838829a7db0e0ad5080d45dc2c545d02aae770a.jpg)  
+Figure 9: Pictorial description of the joint embedding model discussed in the Sec. 5 of the main paper. Modules; $\phi _ { \theta } ( . )$ and $\psi _ { \omega } ( . )$ are implemented as feed-forward neural networks with $P$ and $A \times D$ parameters, respectively. The descriptor of task $k$ $( t ^ { k } )$ is a matrix of dimensions $C _ { k } \times A$ , shared among all the examples of the task, constructed by concatenating the $A$ -dimensional class attribute vectors of $C _ { k }$ classes in the task.

parse/train/Hye-LiR5Y7/Hye-LiR5Y7.md

@@ -0,0 +1,349 @@
+# SOSELETO: A UNIFIED APPROACH TO TRANSFER LEARNING AND TRAINING WITH NOISY LABELS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We present SOSELETO (SOurce SELEction for Target Optimization), a new method for exploiting a source dataset to solve a classification problem on a target dataset. SOSELETO is based on the following simple intuition: some source examples are more informative than others for the target problem. To capture this intuition, source samples are each given weights; these weights are solved for jointly with the source and target classification problems via a bilevel optimization scheme. The target therefore gets to choose the source samples which are most informative for its own classification task. Furthermore, the bilevel nature of the optimization acts as a kind of regularization on the target, mitigating overfitting. SOSELETO may be applied to both classic transfer learning, as well as the problem of training on datasets with noisy labels; we show state of the art results on both of these problems.
+
+![](images/0a13ab3531edf5567f1be60207212bb48226079e1902f06e19754f08c8814355.jpg)  
+Figure 1: SOSELETO applied to a synthetic noisy labels problem. (a) A binary classification problem with points split by the y-axis. Input labels are marked as diamonds and triangles. $2 0 \%$ of the input labels are noisy (have the wrong label). SOSELETO assigns a weight per instance. (b) All correctly labeled input points are assigned high weights. (c) Most noisy points are assigned a low weight. (d) Mean weight of clean and noisy instances throughout training. (e) High accuracy and high recall are achieved for a broad range of weight thresholds.
+
+# 1 INTRODUCTION
+
+Deep learning has made possible many remarkable successes, leading to state of the art algorithms in computer vision, speech and audio, and natural language processing. A key ingredient in this success has been the availability of large datasets. While such datasets are common in certain settings, in other scenarios this is not true. Examples of the latter include “specialist” scenarios, for instance a dataset which is entirely composed of different species of tree; and medical imaging, in which datasets on the order of hundreds to a thousand are common.
+
+A natural question is then how one may apply the techniques of deep learning within these relatively data-poor regimes. A standard approach involves the concept of transfer learning: one uses knowledge gleaned from the source (data-rich regime), and transfers it over to the target (data-poor regime). One of the most common versions of this approach involves a two-stage technique. In the first stage, a network is trained on the source classification task; in the second stage, this network is adapted to the target classification task. There are two variants for this second stage. In feature extraction (e.g. Donahue et al. (2014)), only the parameters of the last layer (i.e. the classifier) are allowed to adapt to the target classification task; whereas in fine-tuning (e.g. Girshick et al. (2014)), the parameters of all of the network layers (i.e. both the features/representation and the classifier) are allowed to adapt. The idea is that by pre-training the network on the source data, a useful feature representation may be learned, which may then be recycled – either partially or completely – for the target regime. This two-stage approach has been quite popular, and works reasonably well on a variety of applications.
+
+Despite this success, we claim that the two-stage approach misses an essential insight: some source examples are more informative than others for the target classification problem. For example, if the source is a large set of natural images and the target consists exclusively of cars, then we might expect that source images of cars, trucks, and motorcycles might be more relevant for the target task than, say, spoons. However, this example is merely illustrative; in practice, the source and target datasets may have no overlapping classes at all. As a result, we don’t know a priori which source examples will be important. Thus, we propose to learn this source filtering as part of an end-to-end training process.
+
+The resulting algorithm is SOSELETO: SOurce SELEction for Target Optimization. Each training sample in the source dataset is given a weight, corresponding to how important it is. The shared source/target representation is then optimized by means of a bilevel optimization. In the interior level, the source minimizes its classification loss with respect to the representation parameters, for fixed values of the sample weights. In the exterior level, the target minimizes its classification loss with respect to both the source sample weights and its own classification layer. The sample weights implicitly control the representation through the interior level. The target therefore gets to choose the source samples which are most informative for its own classification task. Furthermore, the bilevel nature of the optimization acts as a kind of regularization on the target, mitigating overfitting, as the target does not directly control the representation parameters. Finally, note that the entire process – training of the shared representation, target classifier, and source weights – happens simultaneously.
+
+We pause here to note that the general philosophy behind SOSELETO is related to the literature on instance reweighting for domain adaptation, see for example Sugiyama et al. (2008); Yan et al. (2017). However, there is a crucial difference between SOSELETO and this literature, which is related to the difference between domain adaptation and more general transfer learning. Domain adaptation is concerned with the situation in which there is either full overlap between the source and target label sets; or in some more recent work Zhang et al. (2018), partial but significant overlap. Transfer learning, by contrast, refers to the more general situation in which there may be zero overlap between label sets, or possibly very minimal overlap. (For example, if the source consists of natural images and the target of medical images.) The instance reweighting literature is concerned with domain adaptation; the techniques are therefore relevant to the case in which source and target have the same labels. SOSELETO is quite different: it makes no such assumptions, and is therefore a more general approach which can be applied to both “pure” transfer learning, in which there is no overlap between source and target label sets, as well as domain adaptation. (Note also a further distinction with domain adaptation: the target is often – though not always – taken to be unlabelled in domain adaptation. This is not the case for our setting of transfer learning.)
+
+Above, we have illustrated how SOSELETO may be applied to the problem of transfer learning. However, the same algorithm can be applied to the problem of training with noisy labels. Concretely, we assume that there is a large noisy dataset, as well as a much smaller clean dataset; the latter can be constructed cheaply through careful hand-labelling, given its small size. Then if we take the source to be the large noisy dataset, and the target to the small clean dataset, SOSELETO can be applied to the problem. The algorithm will assign high weights to samples with correct labels and low weights to those with incorrect labels, thereby implicitly denoising the source, and allowing for an accurate classifier to be trained.
+
+The remainder of the paper is organized as follows. Section 2 presents related work. Section 3 presents the SOSELETO algorithm, deriving descent equations as well as convergence properties of the bilevel optimization. Section 4 presents results of experiments on both transfer learning as well as training with noisy labels. Section 5 concludes.
+
+# 2 RELATED WORK
+
+Transfer learning As described in Section 1, the most common techniques for transfer learning are feature extraction and fine-tuning, see for example Donahue et al. (2014) and Girshick et al. (2014), respectively. An older survey of transfer learning techniques may be found in Pan & Yang (2010). Domain adaptation Saenko et al. (2010) is concerned with transferring knowledge when the source and target classes are the same. Earlier techniques aligned the source and target via matching of feature space statistics Tzeng et al. (2014); Long et al. (2015); subsequent work used adversarial methods to improve the domain adaptation performance Ganin & Lempitsky (2015); Tzeng et al. (2015; 2017); Hoffman et al. (2017).
+
+In this paper, we are more interested in transfer learning where the source and target classes are different. A series of recent papers Long et al. (2017); Pei et al. (2018); Cao et al. (2018a;b) address domain adaptation that is closer to our setting. In particular, Cao et al. (2018b) examines “partial transfer learning”, the case in which there is partial overlap between source and target classes (particularly when the target classes are a subset of the source). This setting is also dealt with in Busto & Gall (2017).
+
+Ge & Yu (2017) examine the scenario where the source and target classes are completely different. Similar to SOSELETO, they propose selecting a portion of the source dataset. However, the selection is not performed in an end-to-end fashion, as in SOSELETO; rather, selection is performed prior to training, by finding source examples which are similar to the target dataset, where similarity is measured by using filter bank descriptors.
+
+Another recent work of interest is Luo et al. (2017), which focuses on a slightly different scenario: the target consists of a very small number of labelled examples (i.e. the few-shot regime), but a very large number of unlabelled examples. Training is achieved via an adversarial loss to align the source and the target representations, and a special entropy-based loss for the unlabelled part of the data.
+
+Instance reweighting for domain adaptation is a well studied technique, demonstrated e.g. in Covariate Shift methods Shimodaira (2000); Sugiyama et al. (2007; 2008). In these works, the source and target label spaces are the same. We, however, allow for different – even entirely nonoverlapping – classes in the source and target. Crucially, we do not make assumptions on the similarity of the distributions nor do we explicitly optimize for it. The same distinction applies for the recent work of Yan et al. (2017), and for the partial overlap assumption of Zhang et al. (2018). In addition, these two works propose an unsupervised approach, whereas our proposed method is completely supervised. Covariate shift determines the weighting for an instance as the ratio of its probability of being in the training set and being in the prediction set. Consequently, the feature vectors are used in re-weighting, regardless of their labels. This renders covariate shift unsuitable for handling noisy labels. Our re-weighing scheme is instead gradient-based and as we show next performs well in this task.
+
+Learning with noisy labels Classification with noisy labels is a longstanding problem in the machine learning literature, see the review paper Frenay & Verleysen (2014) and the references therein. ´ Within the realm of deep learning, it has been observed that with sufficiently large data, learning with label noise – without modification to the learning algorithms – actually leads to reasonably high accuracy Krause et al. (2016); Sun et al. (2017).
+
+The setting that is of greatest interest to us is when the large noisy dataset is accompanied by a small clean dataset. Sukhbaatar et al. (2014) introduce an additional noise layer into the CNN which attempts to adapt the output to align with the noisy label distribution; the parameters of this layer are also learned. Xiao et al. (2015) use a more general noise model, in which the clean label, noisy label, noise type, and image are jointly specified by a probabilistic graphical model. Both the clean label and the type of noise must be inferred given the image, in this case by two separate CNNs. Li et al. (2017) consider the same setting, but with additional information in the form of a knowledge graph on labels.
+
+Other recent work on label noise includes Rolnick et al. (2017), which shows that adding many copies of an image with noisy labels to a clean dataset barely dents performance; Malach & ShalevShwartz (2017), in which two separate networks are simultaneously trained, and a sample only contributes to the gradient descent step if there is disagreement between the networks (if there is agreement, that probably means the label is wrong); and Drory et al. (2018), which analyzes theoretically the situations in which CNNs are more and less resistant to noise. A pair of papers Liu & Tao (2016); Yu et al. (2017) combine ideas of learning with label noise with instance reweighting.
+
+Bilevel optimization Bilevel optimization problems have a nested structure: the interior level (sometimes called the lower level) is a standard optimization problem; and the exterior level (sometimes called the upper level) is an optimization problem where the objective is a function of the optimal arguments from the interior level. A branch of mathematical programming, bilevel optimization has been extensively studied within this community Colson et al. (2007); Bard (2013). For recent developments, readers are referred to the review paper Sinha et al. (2018). Bilevel optimization has been used in both machine learning, e.g. Bennett et al. (2006; 2008) and computer vision, e.g. Ochs et al. (2015).
+
+# 3 SOSELETO: SOURCE SELECTION FOR TARGET OPTIMIZATION
+
+We have two datasets. The source set is the data-rich set, on which we can learn extensively. It is denoted by $\{ ( x _ { i } ^ { s } , y _ { i } ^ { s } ) \} _ { i = 1 } ^ { n ^ { s } }$ , where as usual $\boldsymbol { x } _ { i } ^ { s }$ is the $i ^ { t h }$ source training image, and $y _ { i } ^ { s }$ is its corresponding label. The second dataset is the target set, which is data-poor; but it is this set which ultimately interests us. That is, the goal in the end is to learn a classifier on the target set, and the source set is only useful insofar as it helps in achieving this goal. The target set is denoted $\{ ( x _ { i } ^ { t } , y _ { i } ^ { t } ) \} _ { i = 1 } ^ { n ^ { t } }$ , and it is assumed that is much smaller than the source set, i.e. $n ^ { t } \ll n ^ { s }$ .
+
+Our goal is to exploit the source set to solve the target classification problem. The key insight is that not all source examples contribute equally useful information in regards to the target problem. For example, suppose that the source set consists of a broad collection of natural images; whereas the target set consists exclusively of various breeds of dog. We would assume that any images of dogs in the source set would help in the target classification task; images of wolves might also help, as might cats. Further afield it might be possible that objects with similar textures as dog fur might be useful, such as rugs. On the flip side, it is probably less likely that images of airplanes and beaches will be relevant (though not impossible). However, the idea is not to come with any preconceived notions (semantic or otherwise) as to which source images will help; rather, the goal is to let the algorithm choose the relevant source images, in an end-to-end fashion.
+
+We assume that the source and target classifier networks have the same architecture, but different network parameters. In particular, the architecture is given by
+
+$$
+F ( x ; \theta , \phi )
+$$
+
+where $\phi$ is last layer, or possibly last few layers, and $\theta$ constitutes all of the remaining layers. We will refer to $\phi$ colloquially as the “classifier”, and to $\theta$ as the “features” or “representation”. (This is consistent with the usage in related papers, see for example Tzeng et al. (2017).) Now, the source and target will share features, but not classifiers; that is, the source network will be given by $F ( x ; \theta , \phi ^ { s } )$ , whereas the target network will be $F ( x ; \theta , \phi ^ { t } )$ . The features $\theta$ are shared between the two, and this is what allows for transfer learning.
+
+The weighted source loss is given by
+
+$$
+L _ { s } ( \theta , \phi ^ { s } , \alpha ) = \frac { 1 } { n ^ { s } } \sum _ { j = 1 } ^ { n ^ { s } } \alpha _ { j } \ell ( y _ { j } ^ { s } , F ( x _ { j } ^ { s } ; \theta , \phi ^ { s } ) )
+$$
+
+where $\alpha _ { j } \in [ 0 , 1 ]$ is a weight assigned to each source training example; and $\ell ( \cdot , \cdot )$ is a per example classification loss, in this case cross-entropy. The use of the weights $\alpha _ { j }$ will allow us to decide which source images are most relevant for the target classification task.
+
+The target loss is standard:
+
+$$
+L _ { t } ( \theta , \phi ^ { t } ) = \frac { 1 } { n ^ { t } } \sum _ { i = 1 } ^ { n ^ { t } } \ell ( y _ { i } ^ { t } , F ( x _ { i } ^ { t } ; \theta , \phi ^ { t } ) )
+$$
+
+As noted in Section 1, this formulation allows us to address both the transfer learning problem as well as learning with label noise. In the former case, the source and target may have non-overlapping label spaces; high weights will indicate which source examples have relevant knowledge for the target classification task. In the latter case, the source is the noisy dataset, the target is the clean dataset, and they share a classifier (i.e. $\phi ^ { t } = \phi ^ { s \cdot }$ ) as well as a label space; high weights will indicate which source examples do not have label noise, and are therefore reliable. In either case, the target is much smaller than the source.
+
+The question now becomes: how can we combine the source and target losses into a single optimization problem? A simple idea is to create a weighted sum of source and target losses. Unfortunately, issues are likely to arise regardless of the weight chosen. If the target is weighted equally to the source, then overfitting may likely result given the small size of the target. On the other hand, if the weights are proportional to the size of the two sets, then the source will simply drown out the target.
+
+A more promising idea is to use bilevel optimization. Specifically, in the interior level we find the optimal features and source classifier as a function of the weights $\alpha$ , by minimizing the source loss:
+
+$$
+\theta ^ { * } ( \alpha ) , \phi ^ { s * } ( \alpha ) = \operatorname* { m i n } _ { \theta , \phi ^ { s } } L _ { s } ( \theta , \phi ^ { s } , \alpha )
+$$
+
+In the exterior level, we minimize the target loss, but only through access to the source weights; that is, we solve:
+
+$$
+\operatorname* { m i n } _ { \alpha , \phi ^ { t } } L _ { t } ( \theta ^ { * } ( \alpha ) , \phi ^ { t } )
+$$
+
+Why might we expect this bilevel formulation to succeed? The key is that the target only has access to the features in an indirect manner, by controlling which source examples are included in the source classification problem. Thus, the target can influence the features chosen, but only in this roundabout way. This serves as an extra form of regularization, mitigating overfitting, which is the main threat when dealing with a small set such as the target.
+
+Implementing the bilevel optimization is rendered somewhat challenging due to the need to solve the optimization problem in the interior level (1). Note that this optimization problem must be solved at every point in time; thus, if we choose to solve the optimization (2) for the exterior level via gradient descent, we will need to solve the interior level optimization (1) at each iteration of the gradient descent. This is clearly inefficient. Furthermore, it is counter to the standard deep learning practice of taking small steps which improve the loss. Thus, we instead propose the following procedure.
+
+At a given iteration, we will take a gradient descent step for the interior level problem (1):
+
+$$
+\begin{array} { c } { { \theta _ { m + 1 } = \theta _ { m } - \lambda _ { p } \frac { \partial L _ { s } } { \partial \theta } ( \theta _ { m } , \phi _ { m } ^ { s } , \alpha _ { m } ) } } \\ { { = \theta _ { m } - \lambda _ { p } Q ( \theta _ { m } , \phi _ { m } ^ { s } ) \alpha _ { m } } } \end{array}
+$$
+
+where $m$ is the iteration number; $\lambda _ { p }$ is the learning rate (where the subscript $p$ stands for “parameters”, to distinguish it from a second learning rate for $\alpha$ , to appear shortly); and $Q ( \theta , \phi ^ { s } )$ is a matrix whose $j ^ { t h }$ column is given by
+
+$$
+q _ { j } = \frac { 1 } { n ^ { s } } \frac { \partial } { \partial \theta } \ell ( y _ { j } ^ { s } , F ( x _ { j } ^ { s } ; \theta , \phi ^ { s } ) )
+$$
+
+Thus, Equation (3) leads to an improvement in the features $\theta$ , for a fixed set of source weights $\alpha$ .   
+Note that there will be an identical descent equation for the classifier $\phi ^ { s }$ , which we omit for clarity.
+
+Given this iterative version of the interior level of the bilevel optimization, we may now turn to the exterior level. Plugging Equation (3) into Equation (2) gives the following problem:
+
+$$
+\operatorname* { m i n } _ { \alpha , \phi ^ { t } } L _ { t } ( \theta _ { m } - \lambda _ { p } Q \alpha , \phi ^ { t } )
+$$
+
+# Algorithm 1 SOSELETO: SOurce SELEction for Target Optimization
+
+where we have suppressed $Q$ ’s arguments for readability. We can then take a gradient descent step of this equation, yielding:
+
+$$
+\begin{array} { l } { \displaystyle \alpha _ { m + 1 } = \alpha _ { m } - \lambda _ { \alpha } \frac { \partial } { \partial \alpha } L _ { t } ( \theta _ { m } - \lambda _ { p } Q \alpha , \phi ^ { t } ) } \\ { \displaystyle = \alpha _ { m } + \lambda _ { \alpha } \lambda _ { p } Q ^ { T } \frac { \partial L _ { t } } { \partial \theta } ( \theta _ { m } - Q \alpha _ { m } \lambda _ { p } ) } \\ { \displaystyle \approx \alpha _ { m } + \lambda _ { \alpha } \lambda _ { p } Q ^ { T } \frac { \partial L _ { t } } { \partial \theta } ( \theta _ { m } ) } \end{array}
+$$
+
+where in the final line, we have made use of the fact that $\lambda _ { p }$ is small. Of course, there will also be a descent equation for the classifier $\phi ^ { t }$ . The resulting update scheme is quite intuitive: source example weights are update according to how well they align with the target aggregated gradient.
+
+We have not yet dealt with the weight constraint. That is, we would like to explicitly require that each $\alpha _ { j } \in [ 0 , 1 ]$ . We may achieve this by requiring $\alpha _ { j } = \sigma ( \beta _ { j } )$ where the new variable $\beta _ { j } \in \mathbb { R }$ , and $\sigma : \mathbb { R }  [ 0 , 1 ]$ is a sigmoid-type function. As shown in Appendix A, for a particular piecewise linear sigmoid function, replacing the Update Equation (4) with a corresponding update equation for $\beta$ is equivalent to modifying Equation (4) to read
+
+$$
+\alpha _ { m + 1 } = \mathbf { C L I P } _ { [ 0 , 1 ] } \left( \alpha _ { m } + \lambda _ { \alpha } \lambda _ { p } Q ^ { T } \frac { \partial L _ { t } } { \partial \theta } ( \theta _ { m } ) \right)
+$$
+
+where $\mathrm { C L I P _ { [ 0 , 1 ] } }$ clips the values below 0 to be $0$ ; and above 1 to be 1.
+
+Thus, SOSELETO consists of alternating Equations (3) and (5), along with the descent equations for the source and target classifiers $\phi ^ { s }$ and $\phi ^ { t }$ . As usual, the whole operation is done on a mini-batch basis, rather than using the entire set; note that if processing is done in parallel, then source minibatches are taken to be non-overlapping, so as to avoid conflicts in the weight updates. SOSELETO is summarized in Algorithm 1. Note that the target derivatives $\partial L _ { t } / \partial \theta$ and $\partial \bar { L _ { t } } / \partial { \phi } ^ { t }$ are evaluated over a target mini-batch; we suppress this for clarity.
+
+In terms of time-complexity, we note that each iteration requires both a source batch and a target batch; assuming identical batch sizes, this means that SOSELETO requires about twice the time as the ordinary source classification problem. Regarding space-complexity, in addition to the ordinary network parameters we need to store the source weights $\alpha$ . Thus, the additional relative spacecomplexity required is the ratio of the source dataset size to the number of network parameters. This is obviously problem and architecture dependent; a typical number might be given by taking the source dataset to be Imagenet ILSVRC-2012 (size 1.2M) and the architecture to be ResNeXt-101 Xie et al. (2017) (size 44.3M parameters), yielding a relative space increase of about $3 \%$ .
+
+Convergence properties SOSELETO is only an approximation to the solution of a bilevel optimization problem. As a result, it is not entirely clear whether it will even converge. In Appendix B, we demonstrate a set of sufficient conditions for SOSELETO to converge to a local minimum of the target loss $L _ { t }$ .
+
+# 4 RESULTS
+
+We briefly discuss some implementation details. In all experiments, we use the SGD optimizer without learning rate decay, and we use $\lambda _ { \alpha } = 1$ . We initialize the $\alpha$ -values to be 1, and in practice clip them to be in the slightly expanded range [0, 1.1]; this allows more relevant source points some room to grow. Other settings are experiment specific, and are discussed in the relevant sections.
+
+# 4.1 NOISY LABELS: SYNTHETIC EXPERIMENT
+
+To illustrate how SOSELETO works on the problem of learning with noisy labels, we begin with a synthetic experiment, see Figure 1. The setting is straightforward: the source dataset consists of 500 points which lie in $\mathbb { R } ^ { 2 }$ . There are two labels / classes, and the ideal separator between the classes is the $y$ -axis. However, of the 500 points, 100 are corrupted: that is, they lie on the wrong side of the separator. This is shown in Figure 1(a), in which one class is shown as white triangles and the second as black pluses. The target dataset is a set of 50 points, which are “clean”, in the sense that they lie on the correct sides of the separator. (For the sake of simplicity, the target set is not illustrated.)
+
+SOSELETO is run for 100 epochs. In Figures 1(b) and 1(c), we choose a threshold of 0.1 on the weights $\alpha$ , and colour the points accordingly. In particular, in Figure 1(b) the clean (i.e. correctly labelled) instances which are above the threshold are labelled in green, while those below the threshold are labelled in red; as can be seen, all of the clean points lie above the threshold for this choice of threshold, meaning that SOSELETO has correctly identified all of the clean points. In Figure 1(c), the noisy (i.e. incorrectly labelled) instances which are below the threshold are labelled in green; and those above the threshold are labelled in red. In this case, SOSELETO correctly identifies most of these noisy labels by assigning them small weights (below 0.1); in fact, 92 out of 100 points are assigned such small weights. The remaining 8 points, those shown in red, are all near the separator, and it is therefore not very surprising that SOSELETO mislabels them. All told, using this particular threshold the algorithm correctly accounts for 492 out of 500 points, i.e. $9 8 . 4 \%$ .
+
+Further analysis appears in Figures 1(d) and 1(e). In Figure 1(e), a plot is shown of mean weight vs. training epoch for clean instances and noisy instances; the width of each plot is the $9 5 \%$ confidence interval of the weights of that type. All weights are initialized at 0.5; after 100 epochs, the clean instances have a mean weight of about 0.8, whereas the noisy instances have a mean weight of about 0.05. The evolution is exactly as one would expect. Figure 1(e) examines the role of the threshold, chose as 0.1 in the above discussion; although 0.1 is a good choice in this case, the good behaviour is fairly robust to choices in the range of 0.1 to 0.4.
+
+# 4.2 NOISY LABELS: CIFAR-10
+
+We now turn to a real-world setting of the problem of learning with label noise. We use a noisy version of CIFAR-10 Krizhevsky & Hinton (2009), following the settings used in Sukhbaatar et al. (2014); Xiao et al. (2015). In particular, an overall noise level is selected. Based on this, a label confusion matrix is chosen such that the diagonal entries of the matrix are equal to one minus the noise level, and the off-diagonals are chosen randomly (while maintaining the matrix’s stochasticity). Noisy labels are then sampled according to this confusion matrix. We run experiments for various overall noise levels.
+
+The target consists of a small clean dataset. CIFAR-10’s train set consists of 50K images; of this 50K, both Sukhbaatar et al. (2014); Xiao et al. (2015) set aside 10K clean examples for pre-training, a necessary step in both of these algorithms. In contrast, we use a smaller clean dataset of half the size, i.e. 5K examples while the rest of the 45K samples are noisy. We compare our results to the two state of the art methods Sukhbaatar et al. (2014); Xiao et al. (2015), as they both address the same setting as we do – the large noisy dataset is accompanied by a small clean dataset, with no extra side-information available. In addition, we compare with the baseline of simply training on the noisy labels without modification. In all cases, Caffes CIFAR-10 Quick cif architecture has been used. For SOSELETO, we use the following settings: $\lambda _ { p } = 1 0 ^ { - 4 }$ , the target batch-size is 32, and the source batch-size is 256. We use a larger source batch-size to enable more $\alpha$ -values to be affected quickly.
+
+Table 1: Noisy labels: CIFAR-10. Best results in bold.   
+
+<table><tr><td>Noise Level</td><td>CIFAR-10 Quick</td><td>Sukhbaatar et al. (2014) 10K clean examples</td><td>Xiao et al. (2015) 10K clean examples</td><td>SOSELETO 5K clean examples</td></tr><tr><td>30%</td><td>65.57</td><td>69.73</td><td>69.81</td><td>72.41</td></tr><tr><td>40%</td><td>62.38</td><td>66.66</td><td>66.76</td><td>69.98</td></tr><tr><td>50%</td><td>57.36</td><td>63.39</td><td>63.00</td><td>66.33</td></tr></table>
+
+Table 2: SVHN 0-4 → MNIST 5-9. Best results in bold.   
+
+<table><tr><td>Uses Unlabelled Data?</td><td>Method</td><td>nt =20</td><td>nt=25</td></tr><tr><td>No</td><td>Target only</td><td>80.1</td><td>84.0</td></tr><tr><td>No</td><td>Fine-tuning</td><td>80.2</td><td>83.0</td></tr><tr><td>No</td><td>SOSELETO</td><td>83.2</td><td>87.9</td></tr><tr><td>Yes</td><td>Vinyals et al. (2016)</td><td>56.6</td><td>51.3</td></tr><tr><td>Yes</td><td>Fine-tuned variant of Vinyals et al. (2016)</td><td>79.3</td><td>82.7</td></tr><tr><td>Yes</td><td>Luo et al. (2017)</td><td>80.4</td><td>83.1</td></tr><tr><td>Yes</td><td>Label-efficient version of Luo et al. (2017)</td><td>94.2</td><td>95.0</td></tr></table>
+
+Results are shown in Table 1 for three different overall noise levels, $30 \%$ , $40 \%$ , and $50 \%$ . Performance is reported for CIFAR-10’s test set, which is of size 10K. (Note that the competitors’ performance numbers are taken from Xiao et al. (2015).) SOSELETO achieves state of the art on all three noise levels, with considerably better performance than both Sukhbaatar et al. (2014) and Xiao et al. (2015): between $2 . 6 \%$ to $3 . 2 \%$ absolute improvement. Furthermore, it does so in each case with only half of the clean samples used in Sukhbaatar et al. (2014); Xiao et al. (2015).
+
+We perform further analysis by examining the $\alpha$ -values that SOSELETO chooses on convergence, see Figure 4.2. To visualize the results, we imagine thresholding the training samples in the source set on the basis of their $\alpha$ -values; we only keep those samples with $\alpha$ greater than a given threshold. By increasing the threshold, we both reduce the total number of samples available, as well as change the effective noise level, which is the fraction of remaining samples which have incorrect labels. We may therefore plot these two quantities against each other, as shown in Figure 4.2; we show three plots, one for each noise level. Looking at these plots, we see for example that for the $3 0 \%$ noise level, if we take the half of the training samples with the highest $\alpha$ -values, we are left with only about $4 \%$ which have incorrect labels. We can therefore see that SOSELETO has effectively filtered out the incorrect labels in this instance. For the $4 0 \%$ and $5 0 \%$ noise levels, the corresponding numbers are about $1 0 \%$ and $2 0 \%$ incorrect labels; while not as effective in the $3 0 \%$ noise level, SOSELETO is still operating as designed. Further evidence for this is provided by the large slopes of all three curves on the righthand side of the graph.
+
+![](images/26975b87d8cb174d50bf1321461eec97ee07f7baf21fd5a4ce1b337bcba7824a.jpg)  
+Figure 2: Noisy labels on CIFAR-10: Effect of $\alpha$ -values chosen by SOSELETO. Blue is $3 0 \%$ noise, green is $4 0 \%$ noise, red is $5 0 \%$ noise. See accompanying explanation in the text.
+
+# 4.3 TRANSFER LEARNING: SVHN 0-4 TO MNIST 5-9
+
+We now examine the performance of SOSELETO on a transfer learning task. In order to provide a challenging setting, we choose to (a) use source and target sets with disjoint label sets, and (b) use a very small target set. In particular, the source dataset is chosen to the subset of Google Street View House Numbers (SVHN) Netzer et al. (2011) corresponding to digits 0-4. SVHN’s train set is of size 73,257 images, with about half of those belonging to the digits 0-4. The target dataset is a very small subset of MNIST LeCun et al. (1998) corresponding to digits 5-9. While MNIST’s train set is of size 60K, with 30K corresponding to digits 5-9, we use very small subsets: either 20 or 25 images, with equal numbers sampled from each class (4 and 5, respectively). Thus, as mentioned, there is no overlap between source and target classes, making it a true transfer learning (rather than domain adaptation) problem; and the small target set size adds further challenge. Furthermore, this task has already been examined in Luo et al. (2017).
+
+We compare our results with the following techniques. Target only, which indicates training on just the target set; standard fine-tuning; Matching Nets Vinyals et al. (2016), a few-shot technique which is relevant given the small target size; fine-tuned Matching Nets, in which the previous result is then fine-tuned on the target set; and two variants of the Label Efficient Learning technique Luo et al. (2017) – one which includes fine-tuning plus a domain adversarial loss, and the other the full technique presented in Luo et al. (2017). Note that besides the target only and fine-tuning approaches, all other approaches depend on unlabelled target data. Specifically, they use all of the remaining MNIST 5-9 examples – about 30,000 – in order to aid in transfer learning. SOSELETO, by contrast, does not make use of any of this data.
+
+For each of the above methods, the simple LeNet architecture LeCun et al. (1998) was used. For SOSELETO, we use the following settings: $\lambda _ { p } = 1 0 ^ { - 2 }$ , the source batch-size is 32, and the target batch-size is 10 (it is chosen to be small since the target itself is very small). Additionally, the SVHN images were resized to $2 8 \times 2 8$ , to match the MNIST size. The performance of the various methods is shown in Table 2, and is reported for MNIST’s test set which is of size 10K. We have divided Table 2 into two parts: those techniques which use the 30K examples of unlabelled data, and those which do not. SOSELETO has superior performance to all of the techniques which do not use unlabelled data. Furthermore, SOSELETO has superior performance to all of the techniques which do use unlabelled data, except the Label Efficient technique. It is noteworthy in particular that SOSELETO outperforms the few-shot techniques, despite not being designed to deal with such small amounts of data.
+
+In Appendix C we further analyze which SVHN instances are considered more useful than others by SOSELETO, by transfering all of SVHN classes to MNSIT 5-9.
+
+Two-stage SOSELETO Finally, we note that although SOSELETO is not designed to use unlabelled data, one may do so using the following two-stage procedure. Stage 1: run SOSELETO as described above. Stage 2: use the learned SOSELETO classifier to classify the unlabelled data. This will now constitute a dataset with noisy labels, and SOSELETO can now be run in the mode of training with label noise, where the noisily labelled unsupervised data is now the source, and the target remains the same small clean set. In the case of $n ^ { t } = 2 5$ , this procedure elevates the accuracy to above ${ \bf 9 2 \% }$ .
+
+# 5 CONCLUSIONS
+
+We have presented SOSELETO, a technique for exploiting a source dataset to learn a target classification task. This exploitation takes the form of joint training through bilevel optimization, in which the source loss is weighted by sample, and is optimized with respect to the network parameters; while the target loss is optimized with respect to these weights and its own classifier. We have derived an efficient algorithm for performing this bilevel optimization, through joint descent in the network parameters and the source weights, and have analyzed the algorithm’s convergence properties. We have empirically shown the effectiveness of the algorithm on both learning with label noise, as well as transfer learning problems. An interesting direction for future research involves incorporating an additional domain alignment term into SOSELETO, in the case where the source and target dataset have overlapping labels. We note that SOSELETO is architecture-agnostic, and thus may be easily deployed. Furthermore, although we have focused on classification tasks, the technique is general and may be applied to other learning tasks within computer vision; this is an important direction for future research.
+
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+
+# APPENDIX A CONSTRAINING THE WEIGHTS
+
+Recall that our goal is to explicitly require that $\alpha _ { j } \in [ 0 , 1 ]$ . We may achieve this by requiring
+
+$$
+\alpha _ { j } = \sigma ( \beta _ { j } ) = \left\{ { \begin{array} { l l } { 0 } & { { \mathrm { i f ~ } } \beta _ { j } < 0 } \\ { \beta _ { j } } & { { \mathrm { i f ~ } } 0 \leq \beta _ { j } \leq 1 } \\ { 1 } & { { \mathrm { i f ~ } } \beta _ { j } > 1 } \end{array} } \right.
+$$
+
+where the new variable $\beta _ { j } \in \mathbb { R }$ , and $\sigma ( \cdot )$ is a kind of piecewise linear sigmoid function.
+
+Now we will wish to replace the Update Equation (4), the update for $\alpha$ , with a corresponding update equation for $\beta$ . This is straightforward. Define the Jacobian $\partial \alpha / \partial \beta$ by
+
+$$
+\left( \frac { \partial \alpha } { \partial \beta } \right) _ { i j } = \frac { \partial \alpha _ { i } } { \partial \beta _ { j } }
+$$
+
+Then we modify Equation (4) to read
+
+$$
+\beta _ { m + 1 } = \beta _ { m } + \lambda _ { \alpha } \lambda _ { p } \left( \frac { \partial \boldsymbol { \alpha } } { \partial \beta } \right) ^ { T } Q ^ { T } \frac { \partial L _ { t } } { \partial \theta } ( \theta _ { m } )
+$$
+
+The Jacobian is easy to compute analytically:
+
+$$
+{ \frac { \partial \alpha } { \partial \beta } } = \mathrm { d i a g } ( \sigma ^ { \prime } ( \beta _ { j } ) ) , \quad { \mathrm { w h e r e } } \quad \sigma ^ { \prime } ( z ) = { \left\{ \begin{array} { l l } { 0 } & { { \mathrm { i f } } \ z < 0 } \\ { 1 } & { { \mathrm { i f } } \ 0 \leq z \leq 1 } \\ { 0 } & { { \mathrm { i f } } \ z > 1 } \end{array} \right. }
+$$
+
+Due to this very simple form, it is easy to see that $\beta _ { m }$ will never lie outside of $[ 0 , 1 ]$ ; and thus that $\alpha _ { m } = \beta _ { m }$ for all time. Hence, we can simply replace this equation with
+
+$$
+\alpha _ { m + 1 } = \mathbf { C L I P } _ { [ 0 , 1 ] } \left( \alpha _ { m } + \lambda _ { \alpha } \lambda _ { p } Q ^ { T } \frac { \partial L _ { t } } { \partial \theta } ( \theta _ { m } ) \right)
+$$
+
+where $\mathrm { C L I P _ { [ 0 , 1 ] } }$ clips the values below 0 to be $0$ ; and above 1 to be 1.
+
+# APPENDIX B PROOF OF CONVERGENCE
+
+SOWETO is only an approximation to the solution of a bilevel optimization problem. As a result, it is not entirely clear whether it will even converge. In this section, we demonstrate a set of sufficient conditions for SOWETO to converge to a local minimum of the target loss $L _ { t }$ .
+
+To this end, let us examine the change in the target loss from iteration $m$ to $m + 1$ :
+
+$$
+\begin{array} { l } { \displaystyle \Delta L _ { t } = L _ { t } ( \theta _ { m + 1 } , \phi _ { m + 1 } ^ { t } ) - L _ { t } ( \theta _ { m } , \phi _ { m } ^ { t } ) } \\ { \displaystyle \quad = L _ { t } \left( \theta _ { m } - \lambda _ { p } Q \alpha _ { m } , \phi _ { m } ^ { t } - \lambda _ { p } \frac { \partial L _ { t } } { \partial \phi ^ { t } } \right) - L _ { t } ( \theta _ { m } , \phi _ { m } ^ { t } ) } \\ { \displaystyle \approx L _ { t } ( \theta _ { m } , \phi _ { m } ^ { t } ) - \lambda _ { p } \left( \frac { \partial L _ { t } } { \partial \theta } \right) ^ { T } Q \alpha _ { m } - \lambda _ { p } \left( \frac { \partial L _ { t } } { \partial \phi ^ { t } } \right) ^ { T } \frac { \partial L _ { t } } { \partial \phi ^ { t } } - L _ { t } ( \theta _ { m } , \phi _ { m } ^ { t } ) } \\ { \displaystyle \quad = - \lambda _ { p } \left( \frac { \partial L _ { t } } { \partial \theta } \right) ^ { T } Q \alpha _ { m } - \lambda _ { p } \left. \frac { \partial L _ { t } } { \partial \phi ^ { t } } \right. ^ { 2 } } \end{array}
+$$
+
+Now, we can use the evolution of the weights $\alpha$ . Specifically, we substitute Equation (4) into the above, to get
+
+$$
+\begin{array} { r } { \Delta L _ { t } \approx - \lambda _ { p } \left( \displaystyle \frac { \partial L _ { t } } { \partial \theta } \right) ^ { T } Q \left( \alpha _ { m - 1 } + \lambda _ { \alpha } \lambda _ { p } Q ^ { T } \displaystyle \frac { \partial L _ { t } } { \partial \theta } \right) - \lambda _ { p } \left\| \displaystyle \frac { \partial L _ { t } } { \partial \phi ^ { t } } \right\| ^ { 2 } } \\ { = - \lambda _ { p } \left( \displaystyle \frac { \partial L _ { t } } { \partial \theta } \right) ^ { T } Q \alpha _ { m - 1 } - \lambda _ { \alpha } \lambda _ { p } ^ { 2 } \left\| Q ^ { T } \displaystyle \frac { \partial L _ { t } } { \partial \theta } \right\| ^ { 2 } - \lambda _ { p } \left\| \displaystyle \frac { \partial L _ { t } } { \partial \phi ^ { t } } \right\| ^ { 2 } } \\ { \equiv \Delta L _ { t } ^ { F O } } \end{array}
+$$
+
+where $\Delta L _ { t } ^ { F O }$ indicates the change in the target loss, to first order.
+
+Note that the latter two terms in $\Delta L _ { t } ^ { F O }$ are both negative, and will therefore cause the first order approximation of the target loss to decrease, as desired. As regards the first term, matters are unclear. However, it is clear that if we set the learning rate $\lambda _ { \alpha }$ sufficiently large, the second term will eventually dominate the first term, and the target loss will be decreased. Indeed, we can do a slightly finer analysis. Ignoring the final term (which is always negative), and setting $\begin{array} { r } { v = Q ^ { T } \frac { \partial L _ { t } } { \partial \theta } } \end{array}$ ∂Lt∂θ , we have that
+
+$$
+\begin{array} { r l } & { \Delta L _ { t } ^ { F O } \leq - \lambda _ { p } v ^ { T } \alpha _ { m - 1 } - \lambda _ { \alpha } \lambda _ { p } ^ { 2 } \| v \| ^ { 2 } } \\ & { \qquad \leq \lambda _ { p } \| v \| _ { 1 } - \lambda _ { \alpha } \lambda _ { p } ^ { 2 } \| v \| _ { 2 } ^ { 2 } } \\ & { \qquad \leq \lambda _ { p } \sqrt { n ^ { s } } \| v \| _ { 2 } - \lambda _ { \alpha } \lambda _ { p } ^ { 2 } \| v \| _ { 2 } ^ { 2 } } \\ & { \qquad = \lambda _ { p } \| v \| _ { 2 } \left( \sqrt { n ^ { s } } - \lambda _ { \alpha } \lambda _ { p } \| v \| _ { 2 } \right) } \end{array}
+$$
+
+where in the second line we have used the fact that all elements of $\alpha$ are in $[ 0 , 1 ]$ ; and in the third line, we have used a standard bound on the $L _ { 1 }$ norm of a vector.
+
+Thus, a sufficient condition for the first order approximation of the target loss to decrease is if
+
+$$
+\lambda _ { \alpha } \geq \frac { \sqrt { n ^ { s } } } { \lambda _ { p } \left. Q ^ { T } \frac { \partial L _ { t } } { \partial \theta } \right. }
+$$
+
+If this is true at all iterations, then the target loss will continually decrease and converge to a local minimum (given that the loss is bounded from below by 0).
+
+# APPENDIX C ANALYZING SVHN 0-9 TO MNIST 5-9
+
+SOSELETO is capable of automatically pruning unhelpful instances at train time. The experiment presented in Section 4.3 demonstrates how SOSELETO can improve classification of MNIST 5-9 by making use of different digits from a different dataset (SVHN 0-4). To further reason about which instances are chosen as useful, we have conducted another experiment: SVHN 0-9 to MNIST 5-9. There is now a partial overlap in classes between source and target. Our findings are summarized in what follows. An immediate effect of increasing the source set, was a dramatic improvement in accuracy to $9 0 . 3 \%$ .
+
+Measuring the percentage of “good” instances (i.e. instances with weight above a certain threshold) didn’t reveal a strong correlation with the labels. In Figure 3 we show this result for a threshold of 0.8. As can be seen, labels 7-9 are slightly higher than the rest but there is no strong evidence of labels 5-9 being more useful than 0-4, as one might hope for.
+
+That said, a more careful examination of low- and high-weighted instances, revealed that the usefulness of an instance, as determined by SOSELETO, is more tied to its appearance: namely, whether the digit is centered, at a similar size as MNIST, the amount of blur, and rotation. In Figure 4 we show a random sample of some “good” and “bad” (i.e. high and low weights, respectively). A close look reveals that “good” instances often tend to be complete, centered, axis aligned, and at a good size (wrt MNIST sizes). Especially interesting was that, among the “bad” instances, we found about $3 - 5 \%$ wrongly labeled instances! In Figure 5 we display several especially problematic instances of the SVHN, all of which are labeled as $\mathbf { \vec { \Delta } } ^ { 6 } 0 ^ { 9 }$ in the dataset. As can be seen, some examples are very clear errors. The highly weighted instances, on the other hand, had almost no such errors.
+
+![](images/ad0e8ef4850d2c76383f216333f07fb753f90f6349b839dd6e38c4cb072114fa.jpg)  
+Figure 3: Percentage of good instances from SVHN per class. Classes 0-4 are colored blue and classes 5-9 are colored orange.
+
+![](images/7f6e513b67f9f7e5bd44df2c81aca3554bedf509af6324444aec81bfd2f4b1f8.jpg)  
+Figure 4: SVHN “good” (left) and “bad” (right) instances of class label 0.
+
+![](images/9f4239d3f182780284611fd6a9e3541da154c4d0fbedfc1a9d2c2fe9d346d466.jpg)  
+Figure 5: Hand-picked examples from the pool of “bad” instances with label 0.

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+        "text": "SOSELETO: A UNIFIED APPROACH TO TRANSFER LEARNING AND TRAINING WITH NOISY LABELS ",
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+        "text": "ABSTRACT ",
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+        "text": "We present SOSELETO (SOurce SELEction for Target Optimization), a new method for exploiting a source dataset to solve a classification problem on a target dataset. SOSELETO is based on the following simple intuition: some source examples are more informative than others for the target problem. To capture this intuition, source samples are each given weights; these weights are solved for jointly with the source and target classification problems via a bilevel optimization scheme. The target therefore gets to choose the source samples which are most informative for its own classification task. Furthermore, the bilevel nature of the optimization acts as a kind of regularization on the target, mitigating overfitting. SOSELETO may be applied to both classic transfer learning, as well as the problem of training on datasets with noisy labels; we show state of the art results on both of these problems. ",
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+        "image_caption": [
+            "Figure 1: SOSELETO applied to a synthetic noisy labels problem. (a) A binary classification problem with points split by the y-axis. Input labels are marked as diamonds and triangles. $2 0 \\%$ of the input labels are noisy (have the wrong label). SOSELETO assigns a weight per instance. (b) All correctly labeled input points are assigned high weights. (c) Most noisy points are assigned a low weight. (d) Mean weight of clean and noisy instances throughout training. (e) High accuracy and high recall are achieved for a broad range of weight thresholds. "
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+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Deep learning has made possible many remarkable successes, leading to state of the art algorithms in computer vision, speech and audio, and natural language processing. A key ingredient in this success has been the availability of large datasets. While such datasets are common in certain settings, in other scenarios this is not true. Examples of the latter include “specialist” scenarios, for instance a dataset which is entirely composed of different species of tree; and medical imaging, in which datasets on the order of hundreds to a thousand are common. ",
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+        "text": "A natural question is then how one may apply the techniques of deep learning within these relatively data-poor regimes. A standard approach involves the concept of transfer learning: one uses knowledge gleaned from the source (data-rich regime), and transfers it over to the target (data-poor regime). One of the most common versions of this approach involves a two-stage technique. In the first stage, a network is trained on the source classification task; in the second stage, this network is adapted to the target classification task. There are two variants for this second stage. In feature extraction (e.g. Donahue et al. (2014)), only the parameters of the last layer (i.e. the classifier) are allowed to adapt to the target classification task; whereas in fine-tuning (e.g. Girshick et al. (2014)), the parameters of all of the network layers (i.e. both the features/representation and the classifier) are allowed to adapt. The idea is that by pre-training the network on the source data, a useful feature representation may be learned, which may then be recycled – either partially or completely – for the target regime. This two-stage approach has been quite popular, and works reasonably well on a variety of applications. ",
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+        "text": "Despite this success, we claim that the two-stage approach misses an essential insight: some source examples are more informative than others for the target classification problem. For example, if the source is a large set of natural images and the target consists exclusively of cars, then we might expect that source images of cars, trucks, and motorcycles might be more relevant for the target task than, say, spoons. However, this example is merely illustrative; in practice, the source and target datasets may have no overlapping classes at all. As a result, we don’t know a priori which source examples will be important. Thus, we propose to learn this source filtering as part of an end-to-end training process. ",
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+        "text": "The resulting algorithm is SOSELETO: SOurce SELEction for Target Optimization. Each training sample in the source dataset is given a weight, corresponding to how important it is. The shared source/target representation is then optimized by means of a bilevel optimization. In the interior level, the source minimizes its classification loss with respect to the representation parameters, for fixed values of the sample weights. In the exterior level, the target minimizes its classification loss with respect to both the source sample weights and its own classification layer. The sample weights implicitly control the representation through the interior level. The target therefore gets to choose the source samples which are most informative for its own classification task. Furthermore, the bilevel nature of the optimization acts as a kind of regularization on the target, mitigating overfitting, as the target does not directly control the representation parameters. Finally, note that the entire process – training of the shared representation, target classifier, and source weights – happens simultaneously. ",
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+        "text": "We pause here to note that the general philosophy behind SOSELETO is related to the literature on instance reweighting for domain adaptation, see for example Sugiyama et al. (2008); Yan et al. (2017). However, there is a crucial difference between SOSELETO and this literature, which is related to the difference between domain adaptation and more general transfer learning. Domain adaptation is concerned with the situation in which there is either full overlap between the source and target label sets; or in some more recent work Zhang et al. (2018), partial but significant overlap. Transfer learning, by contrast, refers to the more general situation in which there may be zero overlap between label sets, or possibly very minimal overlap. (For example, if the source consists of natural images and the target of medical images.) The instance reweighting literature is concerned with domain adaptation; the techniques are therefore relevant to the case in which source and target have the same labels. SOSELETO is quite different: it makes no such assumptions, and is therefore a more general approach which can be applied to both “pure” transfer learning, in which there is no overlap between source and target label sets, as well as domain adaptation. (Note also a further distinction with domain adaptation: the target is often – though not always – taken to be unlabelled in domain adaptation. This is not the case for our setting of transfer learning.) ",
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+        "type": "text",
+        "text": "Above, we have illustrated how SOSELETO may be applied to the problem of transfer learning. However, the same algorithm can be applied to the problem of training with noisy labels. Concretely, we assume that there is a large noisy dataset, as well as a much smaller clean dataset; the latter can be constructed cheaply through careful hand-labelling, given its small size. Then if we take the source to be the large noisy dataset, and the target to the small clean dataset, SOSELETO can be applied to the problem. The algorithm will assign high weights to samples with correct labels and low weights to those with incorrect labels, thereby implicitly denoising the source, and allowing for an accurate classifier to be trained. ",
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+        "text": "The remainder of the paper is organized as follows. Section 2 presents related work. Section 3 presents the SOSELETO algorithm, deriving descent equations as well as convergence properties of the bilevel optimization. Section 4 presents results of experiments on both transfer learning as well as training with noisy labels. Section 5 concludes. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Transfer learning As described in Section 1, the most common techniques for transfer learning are feature extraction and fine-tuning, see for example Donahue et al. (2014) and Girshick et al. (2014), respectively. An older survey of transfer learning techniques may be found in Pan & Yang (2010). Domain adaptation Saenko et al. (2010) is concerned with transferring knowledge when the source and target classes are the same. Earlier techniques aligned the source and target via matching of feature space statistics Tzeng et al. (2014); Long et al. (2015); subsequent work used adversarial methods to improve the domain adaptation performance Ganin & Lempitsky (2015); Tzeng et al. (2015; 2017); Hoffman et al. (2017). ",
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+        "type": "text",
+        "text": "In this paper, we are more interested in transfer learning where the source and target classes are different. A series of recent papers Long et al. (2017); Pei et al. (2018); Cao et al. (2018a;b) address domain adaptation that is closer to our setting. In particular, Cao et al. (2018b) examines “partial transfer learning”, the case in which there is partial overlap between source and target classes (particularly when the target classes are a subset of the source). This setting is also dealt with in Busto & Gall (2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Ge & Yu (2017) examine the scenario where the source and target classes are completely different. Similar to SOSELETO, they propose selecting a portion of the source dataset. However, the selection is not performed in an end-to-end fashion, as in SOSELETO; rather, selection is performed prior to training, by finding source examples which are similar to the target dataset, where similarity is measured by using filter bank descriptors. ",
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+    {
+        "type": "text",
+        "text": "Another recent work of interest is Luo et al. (2017), which focuses on a slightly different scenario: the target consists of a very small number of labelled examples (i.e. the few-shot regime), but a very large number of unlabelled examples. Training is achieved via an adversarial loss to align the source and the target representations, and a special entropy-based loss for the unlabelled part of the data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Instance reweighting for domain adaptation is a well studied technique, demonstrated e.g. in Covariate Shift methods Shimodaira (2000); Sugiyama et al. (2007; 2008). In these works, the source and target label spaces are the same. We, however, allow for different – even entirely nonoverlapping – classes in the source and target. Crucially, we do not make assumptions on the similarity of the distributions nor do we explicitly optimize for it. The same distinction applies for the recent work of Yan et al. (2017), and for the partial overlap assumption of Zhang et al. (2018). In addition, these two works propose an unsupervised approach, whereas our proposed method is completely supervised. Covariate shift determines the weighting for an instance as the ratio of its probability of being in the training set and being in the prediction set. Consequently, the feature vectors are used in re-weighting, regardless of their labels. This renders covariate shift unsuitable for handling noisy labels. Our re-weighing scheme is instead gradient-based and as we show next performs well in this task. ",
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+    {
+        "type": "text",
+        "text": "Learning with noisy labels Classification with noisy labels is a longstanding problem in the machine learning literature, see the review paper Frenay & Verleysen (2014) and the references therein. ´ Within the realm of deep learning, it has been observed that with sufficiently large data, learning with label noise – without modification to the learning algorithms – actually leads to reasonably high accuracy Krause et al. (2016); Sun et al. (2017). ",
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+        "type": "text",
+        "text": "The setting that is of greatest interest to us is when the large noisy dataset is accompanied by a small clean dataset. Sukhbaatar et al. (2014) introduce an additional noise layer into the CNN which attempts to adapt the output to align with the noisy label distribution; the parameters of this layer are also learned. Xiao et al. (2015) use a more general noise model, in which the clean label, noisy label, noise type, and image are jointly specified by a probabilistic graphical model. Both the clean label and the type of noise must be inferred given the image, in this case by two separate CNNs. Li et al. (2017) consider the same setting, but with additional information in the form of a knowledge graph on labels. ",
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+        "type": "text",
+        "text": "Other recent work on label noise includes Rolnick et al. (2017), which shows that adding many copies of an image with noisy labels to a clean dataset barely dents performance; Malach & ShalevShwartz (2017), in which two separate networks are simultaneously trained, and a sample only contributes to the gradient descent step if there is disagreement between the networks (if there is agreement, that probably means the label is wrong); and Drory et al. (2018), which analyzes theoretically the situations in which CNNs are more and less resistant to noise. A pair of papers Liu & Tao (2016); Yu et al. (2017) combine ideas of learning with label noise with instance reweighting. ",
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+        "type": "text",
+        "text": "Bilevel optimization Bilevel optimization problems have a nested structure: the interior level (sometimes called the lower level) is a standard optimization problem; and the exterior level (sometimes called the upper level) is an optimization problem where the objective is a function of the optimal arguments from the interior level. A branch of mathematical programming, bilevel optimization has been extensively studied within this community Colson et al. (2007); Bard (2013). For recent developments, readers are referred to the review paper Sinha et al. (2018). Bilevel optimization has been used in both machine learning, e.g. Bennett et al. (2006; 2008) and computer vision, e.g. Ochs et al. (2015). ",
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+        "type": "text",
+        "text": "3 SOSELETO: SOURCE SELECTION FOR TARGET OPTIMIZATION ",
+        "text_level": 1,
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+        "text": "We have two datasets. The source set is the data-rich set, on which we can learn extensively. It is denoted by $\\{ ( x _ { i } ^ { s } , y _ { i } ^ { s } ) \\} _ { i = 1 } ^ { n ^ { s } }$ , where as usual $\\boldsymbol { x } _ { i } ^ { s }$ is the $i ^ { t h }$ source training image, and $y _ { i } ^ { s }$ is its corresponding label. The second dataset is the target set, which is data-poor; but it is this set which ultimately interests us. That is, the goal in the end is to learn a classifier on the target set, and the source set is only useful insofar as it helps in achieving this goal. The target set is denoted $\\{ ( x _ { i } ^ { t } , y _ { i } ^ { t } ) \\} _ { i = 1 } ^ { n ^ { t } }$ , and it is assumed that is much smaller than the source set, i.e. $n ^ { t } \\ll n ^ { s }$ . ",
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+        "type": "text",
+        "text": "Our goal is to exploit the source set to solve the target classification problem. The key insight is that not all source examples contribute equally useful information in regards to the target problem. For example, suppose that the source set consists of a broad collection of natural images; whereas the target set consists exclusively of various breeds of dog. We would assume that any images of dogs in the source set would help in the target classification task; images of wolves might also help, as might cats. Further afield it might be possible that objects with similar textures as dog fur might be useful, such as rugs. On the flip side, it is probably less likely that images of airplanes and beaches will be relevant (though not impossible). However, the idea is not to come with any preconceived notions (semantic or otherwise) as to which source images will help; rather, the goal is to let the algorithm choose the relevant source images, in an end-to-end fashion. ",
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+        "type": "text",
+        "text": "We assume that the source and target classifier networks have the same architecture, but different network parameters. In particular, the architecture is given by ",
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+        "img_path": "images/b461052d194bd9d2fd0d55fc07bbb2d2077ffad5dae332a1ebf9e26d99ee2628.jpg",
+        "text": "$$\nF ( x ; \\theta , \\phi )\n$$",
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+        "type": "text",
+        "text": "where $\\phi$ is last layer, or possibly last few layers, and $\\theta$ constitutes all of the remaining layers. We will refer to $\\phi$ colloquially as the “classifier”, and to $\\theta$ as the “features” or “representation”. (This is consistent with the usage in related papers, see for example Tzeng et al. (2017).) Now, the source and target will share features, but not classifiers; that is, the source network will be given by $F ( x ; \\theta , \\phi ^ { s } )$ , whereas the target network will be $F ( x ; \\theta , \\phi ^ { t } )$ . The features $\\theta$ are shared between the two, and this is what allows for transfer learning. ",
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+        "type": "text",
+        "text": "The weighted source loss is given by ",
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+        "img_path": "images/0dd1c794ec1ac9641de3509b7d8d38f58a5f1b7722b452dc7ffbcabd0daff6f0.jpg",
+        "text": "$$\nL _ { s } ( \\theta , \\phi ^ { s } , \\alpha ) = \\frac { 1 } { n ^ { s } } \\sum _ { j = 1 } ^ { n ^ { s } } \\alpha _ { j } \\ell ( y _ { j } ^ { s } , F ( x _ { j } ^ { s } ; \\theta , \\phi ^ { s } ) )\n$$",
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+        "type": "text",
+        "text": "where $\\alpha _ { j } \\in [ 0 , 1 ]$ is a weight assigned to each source training example; and $\\ell ( \\cdot , \\cdot )$ is a per example classification loss, in this case cross-entropy. The use of the weights $\\alpha _ { j }$ will allow us to decide which source images are most relevant for the target classification task. ",
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+        "type": "text",
+        "text": "The target loss is standard: ",
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+        "img_path": "images/d6c2c97b12cf94cbbcdae02add811532c8866e5d98731ce5d24c3a098bcd19d4.jpg",
+        "text": "$$\nL _ { t } ( \\theta , \\phi ^ { t } ) = \\frac { 1 } { n ^ { t } } \\sum _ { i = 1 } ^ { n ^ { t } } \\ell ( y _ { i } ^ { t } , F ( x _ { i } ^ { t } ; \\theta , \\phi ^ { t } ) )\n$$",
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+        "type": "text",
+        "text": "As noted in Section 1, this formulation allows us to address both the transfer learning problem as well as learning with label noise. In the former case, the source and target may have non-overlapping label spaces; high weights will indicate which source examples have relevant knowledge for the target classification task. In the latter case, the source is the noisy dataset, the target is the clean dataset, and they share a classifier (i.e. $\\phi ^ { t } = \\phi ^ { s \\cdot }$ ) as well as a label space; high weights will indicate which source examples do not have label noise, and are therefore reliable. In either case, the target is much smaller than the source. ",
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+        "type": "text",
+        "text": "The question now becomes: how can we combine the source and target losses into a single optimization problem? A simple idea is to create a weighted sum of source and target losses. Unfortunately, issues are likely to arise regardless of the weight chosen. If the target is weighted equally to the source, then overfitting may likely result given the small size of the target. On the other hand, if the weights are proportional to the size of the two sets, then the source will simply drown out the target. ",
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+    {
+        "type": "text",
+        "text": "A more promising idea is to use bilevel optimization. Specifically, in the interior level we find the optimal features and source classifier as a function of the weights $\\alpha$ , by minimizing the source loss: ",
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+        "text": "$$\n\\theta ^ { * } ( \\alpha ) , \\phi ^ { s * } ( \\alpha ) = \\operatorname* { m i n } _ { \\theta , \\phi ^ { s } } L _ { s } ( \\theta , \\phi ^ { s } , \\alpha )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "In the exterior level, we minimize the target loss, but only through access to the source weights; that is, we solve: ",
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+        "type": "equation",
+        "img_path": "images/812df57ef828816f80eebd4a6e526644ed9986c9d3997239b8905178d320de97.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\alpha , \\phi ^ { t } } L _ { t } ( \\theta ^ { * } ( \\alpha ) , \\phi ^ { t } )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Why might we expect this bilevel formulation to succeed? The key is that the target only has access to the features in an indirect manner, by controlling which source examples are included in the source classification problem. Thus, the target can influence the features chosen, but only in this roundabout way. This serves as an extra form of regularization, mitigating overfitting, which is the main threat when dealing with a small set such as the target. ",
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+    {
+        "type": "text",
+        "text": "Implementing the bilevel optimization is rendered somewhat challenging due to the need to solve the optimization problem in the interior level (1). Note that this optimization problem must be solved at every point in time; thus, if we choose to solve the optimization (2) for the exterior level via gradient descent, we will need to solve the interior level optimization (1) at each iteration of the gradient descent. This is clearly inefficient. Furthermore, it is counter to the standard deep learning practice of taking small steps which improve the loss. Thus, we instead propose the following procedure. ",
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+    {
+        "type": "text",
+        "text": "At a given iteration, we will take a gradient descent step for the interior level problem (1): ",
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+        "type": "equation",
+        "img_path": "images/349c21e19b98570d3038b0d9b513100fce05d1f0dd4a28057af022769c3eac83.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\theta _ { m + 1 } = \\theta _ { m } - \\lambda _ { p } \\frac { \\partial L _ { s } } { \\partial \\theta } ( \\theta _ { m } , \\phi _ { m } ^ { s } , \\alpha _ { m } ) } } \\\\ { { = \\theta _ { m } - \\lambda _ { p } Q ( \\theta _ { m } , \\phi _ { m } ^ { s } ) \\alpha _ { m } } } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $m$ is the iteration number; $\\lambda _ { p }$ is the learning rate (where the subscript $p$ stands for “parameters”, to distinguish it from a second learning rate for $\\alpha$ , to appear shortly); and $Q ( \\theta , \\phi ^ { s } )$ is a matrix whose $j ^ { t h }$ column is given by ",
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+        "img_path": "images/2b9ac0c9ba956d036f68908e4c1870ddafd56beea52494f5d48101ae9629ed0e.jpg",
+        "text": "$$\nq _ { j } = \\frac { 1 } { n ^ { s } } \\frac { \\partial } { \\partial \\theta } \\ell ( y _ { j } ^ { s } , F ( x _ { j } ^ { s } ; \\theta , \\phi ^ { s } ) )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Thus, Equation (3) leads to an improvement in the features $\\theta$ , for a fixed set of source weights $\\alpha$ .   \nNote that there will be an identical descent equation for the classifier $\\phi ^ { s }$ , which we omit for clarity. ",
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+        "type": "text",
+        "text": "Given this iterative version of the interior level of the bilevel optimization, we may now turn to the exterior level. Plugging Equation (3) into Equation (2) gives the following problem: ",
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+        "text": "$$\n\\operatorname* { m i n } _ { \\alpha , \\phi ^ { t } } L _ { t } ( \\theta _ { m } - \\lambda _ { p } Q \\alpha , \\phi ^ { t } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 SOSELETO: SOurce SELEction for Target Optimization ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where we have suppressed $Q$ ’s arguments for readability. We can then take a gradient descent step of this equation, yielding: ",
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+    {
+        "type": "equation",
+        "img_path": "images/0c1e4c2868a052f857f7e0dc51bdf5eaad6c39ddb184800edec1c4400a130a2b.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\alpha _ { m + 1 } = \\alpha _ { m } - \\lambda _ { \\alpha } \\frac { \\partial } { \\partial \\alpha } L _ { t } ( \\theta _ { m } - \\lambda _ { p } Q \\alpha , \\phi ^ { t } ) } \\\\ { \\displaystyle = \\alpha _ { m } + \\lambda _ { \\alpha } \\lambda _ { p } Q ^ { T } \\frac { \\partial L _ { t } } { \\partial \\theta } ( \\theta _ { m } - Q \\alpha _ { m } \\lambda _ { p } ) } \\\\ { \\displaystyle \\approx \\alpha _ { m } + \\lambda _ { \\alpha } \\lambda _ { p } Q ^ { T } \\frac { \\partial L _ { t } } { \\partial \\theta } ( \\theta _ { m } ) } \\end{array}\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where in the final line, we have made use of the fact that $\\lambda _ { p }$ is small. Of course, there will also be a descent equation for the classifier $\\phi ^ { t }$ . The resulting update scheme is quite intuitive: source example weights are update according to how well they align with the target aggregated gradient. ",
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+    {
+        "type": "text",
+        "text": "We have not yet dealt with the weight constraint. That is, we would like to explicitly require that each $\\alpha _ { j } \\in [ 0 , 1 ]$ . We may achieve this by requiring $\\alpha _ { j } = \\sigma ( \\beta _ { j } )$ where the new variable $\\beta _ { j } \\in \\mathbb { R }$ , and $\\sigma : \\mathbb { R }  [ 0 , 1 ]$ is a sigmoid-type function. As shown in Appendix A, for a particular piecewise linear sigmoid function, replacing the Update Equation (4) with a corresponding update equation for $\\beta$ is equivalent to modifying Equation (4) to read ",
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+    {
+        "type": "equation",
+        "img_path": "images/61fa50399aae5b216dd2f132dd69086acda76fbec15a283dded641a474fd943a.jpg",
+        "text": "$$\n\\alpha _ { m + 1 } = \\mathbf { C L I P } _ { [ 0 , 1 ] } \\left( \\alpha _ { m } + \\lambda _ { \\alpha } \\lambda _ { p } Q ^ { T } \\frac { \\partial L _ { t } } { \\partial \\theta } ( \\theta _ { m } ) \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathrm { C L I P _ { [ 0 , 1 ] } }$ clips the values below 0 to be $0$ ; and above 1 to be 1. ",
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+        "type": "text",
+        "text": "Thus, SOSELETO consists of alternating Equations (3) and (5), along with the descent equations for the source and target classifiers $\\phi ^ { s }$ and $\\phi ^ { t }$ . As usual, the whole operation is done on a mini-batch basis, rather than using the entire set; note that if processing is done in parallel, then source minibatches are taken to be non-overlapping, so as to avoid conflicts in the weight updates. SOSELETO is summarized in Algorithm 1. Note that the target derivatives $\\partial L _ { t } / \\partial \\theta$ and $\\partial \\bar { L _ { t } } / \\partial { \\phi } ^ { t }$ are evaluated over a target mini-batch; we suppress this for clarity. ",
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+        "type": "text",
+        "text": "In terms of time-complexity, we note that each iteration requires both a source batch and a target batch; assuming identical batch sizes, this means that SOSELETO requires about twice the time as the ordinary source classification problem. Regarding space-complexity, in addition to the ordinary network parameters we need to store the source weights $\\alpha$ . Thus, the additional relative spacecomplexity required is the ratio of the source dataset size to the number of network parameters. This is obviously problem and architecture dependent; a typical number might be given by taking the source dataset to be Imagenet ILSVRC-2012 (size 1.2M) and the architecture to be ResNeXt-101 Xie et al. (2017) (size 44.3M parameters), yielding a relative space increase of about $3 \\%$ . ",
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+        "text": "Convergence properties SOSELETO is only an approximation to the solution of a bilevel optimization problem. As a result, it is not entirely clear whether it will even converge. In Appendix B, we demonstrate a set of sufficient conditions for SOSELETO to converge to a local minimum of the target loss $L _ { t }$ . ",
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+        "type": "text",
+        "text": "4 RESULTS ",
+        "text_level": 1,
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+        "text": "We briefly discuss some implementation details. In all experiments, we use the SGD optimizer without learning rate decay, and we use $\\lambda _ { \\alpha } = 1$ . We initialize the $\\alpha$ -values to be 1, and in practice clip them to be in the slightly expanded range [0, 1.1]; this allows more relevant source points some room to grow. Other settings are experiment specific, and are discussed in the relevant sections. ",
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+        "type": "text",
+        "text": "4.1 NOISY LABELS: SYNTHETIC EXPERIMENT ",
+        "text_level": 1,
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+        "text": "To illustrate how SOSELETO works on the problem of learning with noisy labels, we begin with a synthetic experiment, see Figure 1. The setting is straightforward: the source dataset consists of 500 points which lie in $\\mathbb { R } ^ { 2 }$ . There are two labels / classes, and the ideal separator between the classes is the $y$ -axis. However, of the 500 points, 100 are corrupted: that is, they lie on the wrong side of the separator. This is shown in Figure 1(a), in which one class is shown as white triangles and the second as black pluses. The target dataset is a set of 50 points, which are “clean”, in the sense that they lie on the correct sides of the separator. (For the sake of simplicity, the target set is not illustrated.) ",
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+        "type": "text",
+        "text": "SOSELETO is run for 100 epochs. In Figures 1(b) and 1(c), we choose a threshold of 0.1 on the weights $\\alpha$ , and colour the points accordingly. In particular, in Figure 1(b) the clean (i.e. correctly labelled) instances which are above the threshold are labelled in green, while those below the threshold are labelled in red; as can be seen, all of the clean points lie above the threshold for this choice of threshold, meaning that SOSELETO has correctly identified all of the clean points. In Figure 1(c), the noisy (i.e. incorrectly labelled) instances which are below the threshold are labelled in green; and those above the threshold are labelled in red. In this case, SOSELETO correctly identifies most of these noisy labels by assigning them small weights (below 0.1); in fact, 92 out of 100 points are assigned such small weights. The remaining 8 points, those shown in red, are all near the separator, and it is therefore not very surprising that SOSELETO mislabels them. All told, using this particular threshold the algorithm correctly accounts for 492 out of 500 points, i.e. $9 8 . 4 \\%$ . ",
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+        "type": "text",
+        "text": "Further analysis appears in Figures 1(d) and 1(e). In Figure 1(e), a plot is shown of mean weight vs. training epoch for clean instances and noisy instances; the width of each plot is the $9 5 \\%$ confidence interval of the weights of that type. All weights are initialized at 0.5; after 100 epochs, the clean instances have a mean weight of about 0.8, whereas the noisy instances have a mean weight of about 0.05. The evolution is exactly as one would expect. Figure 1(e) examines the role of the threshold, chose as 0.1 in the above discussion; although 0.1 is a good choice in this case, the good behaviour is fairly robust to choices in the range of 0.1 to 0.4. ",
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+    {
+        "type": "text",
+        "text": "4.2 NOISY LABELS: CIFAR-10 ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We now turn to a real-world setting of the problem of learning with label noise. We use a noisy version of CIFAR-10 Krizhevsky & Hinton (2009), following the settings used in Sukhbaatar et al. (2014); Xiao et al. (2015). In particular, an overall noise level is selected. Based on this, a label confusion matrix is chosen such that the diagonal entries of the matrix are equal to one minus the noise level, and the off-diagonals are chosen randomly (while maintaining the matrix’s stochasticity). Noisy labels are then sampled according to this confusion matrix. We run experiments for various overall noise levels. ",
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+        "type": "text",
+        "text": "The target consists of a small clean dataset. CIFAR-10’s train set consists of 50K images; of this 50K, both Sukhbaatar et al. (2014); Xiao et al. (2015) set aside 10K clean examples for pre-training, a necessary step in both of these algorithms. In contrast, we use a smaller clean dataset of half the size, i.e. 5K examples while the rest of the 45K samples are noisy. We compare our results to the two state of the art methods Sukhbaatar et al. (2014); Xiao et al. (2015), as they both address the same setting as we do – the large noisy dataset is accompanied by a small clean dataset, with no extra side-information available. In addition, we compare with the baseline of simply training on the noisy labels without modification. In all cases, Caffes CIFAR-10 Quick cif architecture has been used. For SOSELETO, we use the following settings: $\\lambda _ { p } = 1 0 ^ { - 4 }$ , the target batch-size is 32, and the source batch-size is 256. We use a larger source batch-size to enable more $\\alpha$ -values to be affected quickly. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/0346c51b76450760f2502e92de7942afdf44713fc2cd71c04ac78593e78a2f48.jpg",
+        "table_caption": [
+            "Table 1: Noisy labels: CIFAR-10. Best results in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Noise Level</td><td>CIFAR-10 Quick</td><td>Sukhbaatar et al. (2014) 10K clean examples</td><td>Xiao et al. (2015) 10K clean examples</td><td>SOSELETO 5K clean examples</td></tr><tr><td>30%</td><td>65.57</td><td>69.73</td><td>69.81</td><td>72.41</td></tr><tr><td>40%</td><td>62.38</td><td>66.66</td><td>66.76</td><td>69.98</td></tr><tr><td>50%</td><td>57.36</td><td>63.39</td><td>63.00</td><td>66.33</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ebff97e24b654ee3a471a696a62397d5b91ce3334857586619f9d0b0aa410614.jpg",
+        "table_caption": [
+            "Table 2: SVHN 0-4 → MNIST 5-9. Best results in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Uses Unlabelled Data?</td><td>Method</td><td>nt =20</td><td>nt=25</td></tr><tr><td>No</td><td>Target only</td><td>80.1</td><td>84.0</td></tr><tr><td>No</td><td>Fine-tuning</td><td>80.2</td><td>83.0</td></tr><tr><td>No</td><td>SOSELETO</td><td>83.2</td><td>87.9</td></tr><tr><td>Yes</td><td>Vinyals et al. (2016)</td><td>56.6</td><td>51.3</td></tr><tr><td>Yes</td><td>Fine-tuned variant of Vinyals et al. (2016)</td><td>79.3</td><td>82.7</td></tr><tr><td>Yes</td><td>Luo et al. (2017)</td><td>80.4</td><td>83.1</td></tr><tr><td>Yes</td><td>Label-efficient version of Luo et al. (2017)</td><td>94.2</td><td>95.0</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results are shown in Table 1 for three different overall noise levels, $30 \\%$ , $40 \\%$ , and $50 \\%$ . Performance is reported for CIFAR-10’s test set, which is of size 10K. (Note that the competitors’ performance numbers are taken from Xiao et al. (2015).) SOSELETO achieves state of the art on all three noise levels, with considerably better performance than both Sukhbaatar et al. (2014) and Xiao et al. (2015): between $2 . 6 \\%$ to $3 . 2 \\%$ absolute improvement. Furthermore, it does so in each case with only half of the clean samples used in Sukhbaatar et al. (2014); Xiao et al. (2015). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We perform further analysis by examining the $\\alpha$ -values that SOSELETO chooses on convergence, see Figure 4.2. To visualize the results, we imagine thresholding the training samples in the source set on the basis of their $\\alpha$ -values; we only keep those samples with $\\alpha$ greater than a given threshold. By increasing the threshold, we both reduce the total number of samples available, as well as change the effective noise level, which is the fraction of remaining samples which have incorrect labels. We may therefore plot these two quantities against each other, as shown in Figure 4.2; we show three plots, one for each noise level. Looking at these plots, we see for example that for the $3 0 \\%$ noise level, if we take the half of the training samples with the highest $\\alpha$ -values, we are left with only about $4 \\%$ which have incorrect labels. We can therefore see that SOSELETO has effectively filtered out the incorrect labels in this instance. For the $4 0 \\%$ and $5 0 \\%$ noise levels, the corresponding numbers are about $1 0 \\%$ and $2 0 \\%$ incorrect labels; while not as effective in the $3 0 \\%$ noise level, SOSELETO is still operating as designed. Further evidence for this is provided by the large slopes of all three curves on the righthand side of the graph. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/26975b87d8cb174d50bf1321461eec97ee07f7baf21fd5a4ce1b337bcba7824a.jpg",
+        "image_caption": [
+            "Figure 2: Noisy labels on CIFAR-10: Effect of $\\alpha$ -values chosen by SOSELETO. Blue is $3 0 \\%$ noise, green is $4 0 \\%$ noise, red is $5 0 \\%$ noise. See accompanying explanation in the text. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.3 TRANSFER LEARNING: SVHN 0-4 TO MNIST 5-9 ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We now examine the performance of SOSELETO on a transfer learning task. In order to provide a challenging setting, we choose to (a) use source and target sets with disjoint label sets, and (b) use a very small target set. In particular, the source dataset is chosen to the subset of Google Street View House Numbers (SVHN) Netzer et al. (2011) corresponding to digits 0-4. SVHN’s train set is of size 73,257 images, with about half of those belonging to the digits 0-4. The target dataset is a very small subset of MNIST LeCun et al. (1998) corresponding to digits 5-9. While MNIST’s train set is of size 60K, with 30K corresponding to digits 5-9, we use very small subsets: either 20 or 25 images, with equal numbers sampled from each class (4 and 5, respectively). Thus, as mentioned, there is no overlap between source and target classes, making it a true transfer learning (rather than domain adaptation) problem; and the small target set size adds further challenge. Furthermore, this task has already been examined in Luo et al. (2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare our results with the following techniques. Target only, which indicates training on just the target set; standard fine-tuning; Matching Nets Vinyals et al. (2016), a few-shot technique which is relevant given the small target size; fine-tuned Matching Nets, in which the previous result is then fine-tuned on the target set; and two variants of the Label Efficient Learning technique Luo et al. (2017) – one which includes fine-tuning plus a domain adversarial loss, and the other the full technique presented in Luo et al. (2017). Note that besides the target only and fine-tuning approaches, all other approaches depend on unlabelled target data. Specifically, they use all of the remaining MNIST 5-9 examples – about 30,000 – in order to aid in transfer learning. SOSELETO, by contrast, does not make use of any of this data. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For each of the above methods, the simple LeNet architecture LeCun et al. (1998) was used. For SOSELETO, we use the following settings: $\\lambda _ { p } = 1 0 ^ { - 2 }$ , the source batch-size is 32, and the target batch-size is 10 (it is chosen to be small since the target itself is very small). Additionally, the SVHN images were resized to $2 8 \\times 2 8$ , to match the MNIST size. The performance of the various methods is shown in Table 2, and is reported for MNIST’s test set which is of size 10K. We have divided Table 2 into two parts: those techniques which use the 30K examples of unlabelled data, and those which do not. SOSELETO has superior performance to all of the techniques which do not use unlabelled data. Furthermore, SOSELETO has superior performance to all of the techniques which do use unlabelled data, except the Label Efficient technique. It is noteworthy in particular that SOSELETO outperforms the few-shot techniques, despite not being designed to deal with such small amounts of data. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "In Appendix C we further analyze which SVHN instances are considered more useful than others by SOSELETO, by transfering all of SVHN classes to MNSIT 5-9. ",
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+    {
+        "type": "text",
+        "text": "Two-stage SOSELETO Finally, we note that although SOSELETO is not designed to use unlabelled data, one may do so using the following two-stage procedure. Stage 1: run SOSELETO as described above. Stage 2: use the learned SOSELETO classifier to classify the unlabelled data. This will now constitute a dataset with noisy labels, and SOSELETO can now be run in the mode of training with label noise, where the noisily labelled unsupervised data is now the source, and the target remains the same small clean set. In the case of $n ^ { t } = 2 5$ , this procedure elevates the accuracy to above ${ \\bf 9 2 \\% }$ . ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We have presented SOSELETO, a technique for exploiting a source dataset to learn a target classification task. This exploitation takes the form of joint training through bilevel optimization, in which the source loss is weighted by sample, and is optimized with respect to the network parameters; while the target loss is optimized with respect to these weights and its own classifier. We have derived an efficient algorithm for performing this bilevel optimization, through joint descent in the network parameters and the source weights, and have analyzed the algorithm’s convergence properties. We have empirically shown the effectiveness of the algorithm on both learning with label noise, as well as transfer learning problems. An interesting direction for future research involves incorporating an additional domain alignment term into SOSELETO, in the case where the source and target dataset have overlapping labels. We note that SOSELETO is architecture-agnostic, and thus may be easily deployed. Furthermore, although we have focused on classification tasks, the technique is general and may be applied to other learning tasks within computer vision; this is an important direction for future research. ",
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+        "text": "",
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+    {
+        "type": "text",
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+    },
+    {
+        "type": "text",
+        "text": "APPENDIX A CONSTRAINING THE WEIGHTS",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Recall that our goal is to explicitly require that $\\alpha _ { j } \\in [ 0 , 1 ]$ . We may achieve this by requiring ",
+        "bbox": [
+            173,
+            132,
+            781,
+            148
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/841b0f72e5ae8ca8fb7006e8beb7996bcd06f4c3c27eeceb6a46ffd1266a2448.jpg",
+        "text": "$$\n\\alpha _ { j } = \\sigma ( \\beta _ { j } ) = \\left\\{ { \\begin{array} { l l } { 0 } & { { \\mathrm { i f ~ } } \\beta _ { j } < 0 } \\\\ { \\beta _ { j } } & { { \\mathrm { i f ~ } } 0 \\leq \\beta _ { j } \\leq 1 } \\\\ { 1 } & { { \\mathrm { i f ~ } } \\beta _ { j } > 1 } \\end{array} } \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            155,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where the new variable $\\beta _ { j } \\in \\mathbb { R }$ , and $\\sigma ( \\cdot )$ is a kind of piecewise linear sigmoid function. ",
+        "bbox": [
+            173,
+            212,
+            746,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Now we will wish to replace the Update Equation (4), the update for $\\alpha$ , with a corresponding update equation for $\\beta$ . This is straightforward. Define the Jacobian $\\partial \\alpha / \\partial \\beta$ by ",
+        "bbox": [
+            173,
+            233,
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+            262
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/02c8a8148120ee5c371f473e2714688e2e41c02add4bb5999bcaca6a50420291.jpg",
+        "text": "$$\n\\left( \\frac { \\partial \\alpha } { \\partial \\beta } \\right) _ { i j } = \\frac { \\partial \\alpha _ { i } } { \\partial \\beta _ { j } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            441,
+            266,
+            557,
+            304
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Then we modify Equation (4) to read ",
+        "bbox": [
+            174,
+            308,
+            419,
+            323
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/60f111c6b64585d6d58fc05d21394feb51faaa2fbf002ab1b9762f8eb4d570cb.jpg",
+        "text": "$$\n\\beta _ { m + 1 } = \\beta _ { m } + \\lambda _ { \\alpha } \\lambda _ { p } \\left( \\frac { \\partial \\boldsymbol { \\alpha } } { \\partial \\beta } \\right) ^ { T } Q ^ { T } \\frac { \\partial L _ { t } } { \\partial \\theta } ( \\theta _ { m } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            327,
+            645,
+            364
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The Jacobian is easy to compute analytically: ",
+        "bbox": [
+            174,
+            376,
+            472,
+            391
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1d908bcd0de1e48b049fa8fce021b2993e8704e32c0079948417e85a6c0111da.jpg",
+        "text": "$$\n{ \\frac { \\partial \\alpha } { \\partial \\beta } } = \\mathrm { d i a g } ( \\sigma ^ { \\prime } ( \\beta _ { j } ) ) , \\quad { \\mathrm { w h e r e } } \\quad \\sigma ^ { \\prime } ( z ) = { \\left\\{ \\begin{array} { l l } { 0 } & { { \\mathrm { i f } } \\ z < 0 } \\\\ { 1 } & { { \\mathrm { i f } } \\ 0 \\leq z \\leq 1 } \\\\ { 0 } & { { \\mathrm { i f } } \\ z > 1 } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            396,
+            694,
+            448
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Due to this very simple form, it is easy to see that $\\beta _ { m }$ will never lie outside of $[ 0 , 1 ]$ ; and thus that $\\alpha _ { m } = \\beta _ { m }$ for all time. Hence, we can simply replace this equation with ",
+        "bbox": [
+            173,
+            454,
+            823,
+            483
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1befc401a25fdc81e716a13b5d341e2deedcf0ba7c6e4f70d763eb4200164e65.jpg",
+        "text": "$$\n\\alpha _ { m + 1 } = \\mathbf { C L I P } _ { [ 0 , 1 ] } \\left( \\alpha _ { m } + \\lambda _ { \\alpha } \\lambda _ { p } Q ^ { T } \\frac { \\partial L _ { t } } { \\partial \\theta } ( \\theta _ { m } ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            487,
+            656,
+            522
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { C L I P _ { [ 0 , 1 ] } }$ clips the values below 0 to be $0$ ; and above 1 to be 1. ",
+        "bbox": [
+            176,
+            525,
+            627,
+            541
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX B PROOF OF CONVERGENCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            559,
+            522,
+            575
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "SOWETO is only an approximation to the solution of a bilevel optimization problem. As a result, it is not entirely clear whether it will even converge. In this section, we demonstrate a set of sufficient conditions for SOWETO to converge to a local minimum of the target loss $L _ { t }$ . ",
+        "bbox": [
+            173,
+            590,
+            825,
+            633
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To this end, let us examine the change in the target loss from iteration $m$ to $m + 1$ : ",
+        "bbox": [
+            173,
+            640,
+            715,
+            655
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/98452226701a0c8ecf974fe86a8b3e7989a8a4c4368604bc1842f99110beb7c9.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\Delta L _ { t } = L _ { t } ( \\theta _ { m + 1 } , \\phi _ { m + 1 } ^ { t } ) - L _ { t } ( \\theta _ { m } , \\phi _ { m } ^ { t } ) } \\\\ { \\displaystyle \\quad = L _ { t } \\left( \\theta _ { m } - \\lambda _ { p } Q \\alpha _ { m } , \\phi _ { m } ^ { t } - \\lambda _ { p } \\frac { \\partial L _ { t } } { \\partial \\phi ^ { t } } \\right) - L _ { t } ( \\theta _ { m } , \\phi _ { m } ^ { t } ) } \\\\ { \\displaystyle \\approx L _ { t } ( \\theta _ { m } , \\phi _ { m } ^ { t } ) - \\lambda _ { p } \\left( \\frac { \\partial L _ { t } } { \\partial \\theta } \\right) ^ { T } Q \\alpha _ { m } - \\lambda _ { p } \\left( \\frac { \\partial L _ { t } } { \\partial \\phi ^ { t } } \\right) ^ { T } \\frac { \\partial L _ { t } } { \\partial \\phi ^ { t } } - L _ { t } ( \\theta _ { m } , \\phi _ { m } ^ { t } ) } \\\\ { \\displaystyle \\quad = - \\lambda _ { p } \\left( \\frac { \\partial L _ { t } } { \\partial \\theta } \\right) ^ { T } Q \\alpha _ { m } - \\lambda _ { p } \\left. \\frac { \\partial L _ { t } } { \\partial \\phi ^ { t } } \\right. ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            659,
+            756,
+            790
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Now, we can use the evolution of the weights $\\alpha$ . Specifically, we substitute Equation (4) into the above, to get ",
+        "bbox": [
+            173,
+            792,
+            825,
+            821
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/628e4976b5f978639395097d2193bcc090963508cb61523c82af2c8622ef8eda.jpg",
+        "text": "$$\n\\begin{array} { r } { \\Delta L _ { t } \\approx - \\lambda _ { p } \\left( \\displaystyle \\frac { \\partial L _ { t } } { \\partial \\theta } \\right) ^ { T } Q \\left( \\alpha _ { m - 1 } + \\lambda _ { \\alpha } \\lambda _ { p } Q ^ { T } \\displaystyle \\frac { \\partial L _ { t } } { \\partial \\theta } \\right) - \\lambda _ { p } \\left\\| \\displaystyle \\frac { \\partial L _ { t } } { \\partial \\phi ^ { t } } \\right\\| ^ { 2 } } \\\\ { = - \\lambda _ { p } \\left( \\displaystyle \\frac { \\partial L _ { t } } { \\partial \\theta } \\right) ^ { T } Q \\alpha _ { m - 1 } - \\lambda _ { \\alpha } \\lambda _ { p } ^ { 2 } \\left\\| Q ^ { T } \\displaystyle \\frac { \\partial L _ { t } } { \\partial \\theta } \\right\\| ^ { 2 } - \\lambda _ { p } \\left\\| \\displaystyle \\frac { \\partial L _ { t } } { \\partial \\phi ^ { t } } \\right\\| ^ { 2 } } \\\\ { \\equiv \\Delta L _ { t } ^ { F O } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            277,
+            825,
+            718,
+            922
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $\\Delta L _ { t } ^ { F O }$ indicates the change in the target loss, to first order. ",
+        "bbox": [
+            174,
+            102,
+            604,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Note that the latter two terms in $\\Delta L _ { t } ^ { F O }$ are both negative, and will therefore cause the first order approximation of the target loss to decrease, as desired. As regards the first term, matters are unclear. However, it is clear that if we set the learning rate $\\lambda _ { \\alpha }$ sufficiently large, the second term will eventually dominate the first term, and the target loss will be decreased. Indeed, we can do a slightly finer analysis. Ignoring the final term (which is always negative), and setting $\\begin{array} { r } { v = Q ^ { T } \\frac { \\partial L _ { t } } { \\partial \\theta } } \\end{array}$ ∂Lt∂θ , we have that ",
+        "bbox": [
+            173,
+            123,
+            825,
+            210
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7b1aa308fb6692b50d0e19ed5d6c5fb81ae249ede16b7007aebc7f81e56a5f8a.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\Delta L _ { t } ^ { F O } \\leq - \\lambda _ { p } v ^ { T } \\alpha _ { m - 1 } - \\lambda _ { \\alpha } \\lambda _ { p } ^ { 2 } \\| v \\| ^ { 2 } } \\\\ & { \\qquad \\leq \\lambda _ { p } \\| v \\| _ { 1 } - \\lambda _ { \\alpha } \\lambda _ { p } ^ { 2 } \\| v \\| _ { 2 } ^ { 2 } } \\\\ & { \\qquad \\leq \\lambda _ { p } \\sqrt { n ^ { s } } \\| v \\| _ { 2 } - \\lambda _ { \\alpha } \\lambda _ { p } ^ { 2 } \\| v \\| _ { 2 } ^ { 2 } } \\\\ & { \\qquad = \\lambda _ { p } \\| v \\| _ { 2 } \\left( \\sqrt { n ^ { s } } - \\lambda _ { \\alpha } \\lambda _ { p } \\| v \\| _ { 2 } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            214,
+            630,
+            308
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where in the second line we have used the fact that all elements of $\\alpha$ are in $[ 0 , 1 ]$ ; and in the third line, we have used a standard bound on the $L _ { 1 }$ norm of a vector. ",
+        "bbox": [
+            171,
+            330,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Thus, a sufficient condition for the first order approximation of the target loss to decrease is if ",
+        "bbox": [
+            174,
+            366,
+            789,
+            381
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7d0582dcc84c0490e9bf11d75f7af9e2d9d6332222041fa4e786ee5687573c41.jpg",
+        "text": "$$\n\\lambda _ { \\alpha } \\geq \\frac { \\sqrt { n ^ { s } } } { \\lambda _ { p } \\left. Q ^ { T } \\frac { \\partial L _ { t } } { \\partial \\theta } \\right. }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            431,
+            386,
+            566,
+            426
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "If this is true at all iterations, then the target loss will continually decrease and converge to a local minimum (given that the loss is bounded from below by 0). ",
+        "bbox": [
+            173,
+            431,
+            823,
+            459
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX C ANALYZING SVHN 0-9 TO MNIST 5-9 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            478,
+            632,
+            496
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "SOSELETO is capable of automatically pruning unhelpful instances at train time. The experiment presented in Section 4.3 demonstrates how SOSELETO can improve classification of MNIST 5-9 by making use of different digits from a different dataset (SVHN 0-4). To further reason about which instances are chosen as useful, we have conducted another experiment: SVHN 0-9 to MNIST 5-9. There is now a partial overlap in classes between source and target. Our findings are summarized in what follows. An immediate effect of increasing the source set, was a dramatic improvement in accuracy to $9 0 . 3 \\%$ . ",
+        "bbox": [
+            173,
+            510,
+            825,
+            608
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Measuring the percentage of “good” instances (i.e. instances with weight above a certain threshold) didn’t reveal a strong correlation with the labels. In Figure 3 we show this result for a threshold of 0.8. As can be seen, labels 7-9 are slightly higher than the rest but there is no strong evidence of labels 5-9 being more useful than 0-4, as one might hope for. ",
+        "bbox": [
+            174,
+            614,
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+            671
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "That said, a more careful examination of low- and high-weighted instances, revealed that the usefulness of an instance, as determined by SOSELETO, is more tied to its appearance: namely, whether the digit is centered, at a similar size as MNIST, the amount of blur, and rotation. In Figure 4 we show a random sample of some “good” and “bad” (i.e. high and low weights, respectively). A close look reveals that “good” instances often tend to be complete, centered, axis aligned, and at a good size (wrt MNIST sizes). Especially interesting was that, among the “bad” instances, we found about $3 - 5 \\%$ wrongly labeled instances! In Figure 5 we display several especially problematic instances of the SVHN, all of which are labeled as $\\mathbf { \\vec { \\Delta } } ^ { 6 } 0 ^ { 9 }$ in the dataset. As can be seen, some examples are very clear errors. The highly weighted instances, on the other hand, had almost no such errors. ",
+        "bbox": [
+            173,
+            678,
+            825,
+            804
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/ad0e8ef4850d2c76383f216333f07fb753f90f6349b839dd6e38c4cb072114fa.jpg",
+        "image_caption": [
+            "Figure 3: Percentage of good instances from SVHN per class. Classes 0-4 are colored blue and classes 5-9 are colored orange. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            349,
+            142,
+            630,
+            313
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/7f6e513b67f9f7e5bd44df2c81aca3554bedf509af6324444aec81bfd2f4b1f8.jpg",
+        "image_caption": [
+            "Figure 4: SVHN “good” (left) and “bad” (right) instances of class label 0. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            434,
+            841,
+            574
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/9f4239d3f182780284611fd6a9e3541da154c4d0fbedfc1a9d2c2fe9d346d466.jpg",
+        "image_caption": [
+            "Figure 5: Hand-picked examples from the pool of “bad” instances with label 0. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            250,
+            695,
+            745,
+            848
+        ],
+        "page_idx": 15
+    }
+]

parse/train/Hyepjh4FwB/Hyepjh4FwB.md

@@ -0,0 +1,454 @@
+# PROTOATTEND: ATTENTION-BASED PROTOTYPICAL LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We propose a novel inherently interpretable machine learning method that bases decisions on few relevant examples that we call prototypes. Our method, ProtoAttend, can be integrated into a wide range of neural network architectures including pre-trained models. It utilizes an attention mechanism that relates the encoded representations to samples in order to determine prototypes. The resulting model outperforms state of the art in three high impact problems without sacrificing accuracy of the original model: (1) it enables high-quality interpretability that outputs samples most relevant to the decision-making (i.e. a sample-based interpretability method); (2) it achieves state of the art confidence estimation by quantifying the mismatch across prototype labels; and (3) it obtains state of the art in distribution mismatch detection. All this can be achieved with minimal additional test time and a practically viable training time computational cost.
+
+# 1 INTRODUCTION
+
+Deep neural networks have been pushing the frontiers of artificial intelligence (AI) by yielding excellent performance in numerous tasks, from understanding images (He et al., 2016) to text (Conneau et al., 2016). Yet, high performance is not always a sufficient factor - as some realworld deployment scenarios might necessitate that an ideal AI system is ‘interpretable’, such that it builds trust by explaining rationales behind decisions, allow detection of common failure cases and biases, and refrains from making decisions without sufficient confidence. In their conventional form, deep neural networks are considered as black-box models – they are controlled by complex nonlinear interactions between many parameters that are difficult to understand. There are numerous approaches, (Kim et al., 2018; Erhan et al., 2009; Zeiler & Fergus, 2013; Simonyan et al., 2013), that bring post-hoc explainability of decisions to already-trained models. Yet, these have the fundamental limitation that the models are not designed for interpretability. There are also approaches on the redesign of neural networks towards making them inherently-interpretable, as in this paper. Some notable ones include sequential attention (Bahdanau et al., 2015), capsule networks (Sabour et al., 2017), and interpretable convolutional filters (Zhang et al., 2018).
+
+![](images/141a9afc992c57a8503608a14f6d6de9425d4c43ba3c5f51d12692c086a72674.jpg)  
+Figure 1: ProtoAttend bases the decision on a few prototypes from the database. This enables interpretability of the prediction (by visualizing the highest weight prototypes) and confidence estimation for the decision (by measuring agreement across prototype labels).
+
+We focus on inherently-interpretable deep neural network modeling with the foundations of prototypical learning. Prototypical learning decomposes decision making into known samples (see Fig. 1), referred here as prototypes. We base our method on the principle that prototypes should constitute a minimal subset of samples with high interpretable value that can serve as a distillation or condensed view of a dataset (Bien & Tibshirani, 2012). Given that the number of objects a human can interpret is limited (Miller, 1956), outputting few prototypes can be an effective approach for humans to understand the AI model behavior. In addition to such interpretability, prototypical learning:
+
+(1) provides an efficient confidence metric by measuring mismatches in prototype labels, allowing performance to be improved by refraining from making predictions in the absence of sufficient confidence, (2) helps detect deviations in the test distribution by measuring mismatches in prototype labels that represent the support of the training dataset, and (3) enables performance in the high label noise regime to be improved by controlling the number of selected prototypes. Given these motivations, prototypes should be controllable in number, and should be perceptually relevant to the input in explaining the decision making task. Prototype selection in its naive form is computationally expensive and perceptually challenging (Bien & Tibshirani, 2012). We design ProtoAttend to address this problem in an efficient way. Our contributions can be summarized as follows:
+
+1. We present principles that can be guiding for the design of inherently-interpretable models based on sample-based interpretability.   
+2. We propose a novel method, ProtoAttend, for selecting input-dependent prototypes based on an attention mechanism between the input and prototype candidates. ProtoAttend is model-agnostic and can even be integrated with pre-trained models.   
+3. ProtoAttend allows interpreting the contribution of each prototype via the attention outputs.   
+4. For a ‘condensed view’, we demonstrate that sparsity in weights can be efficiently imposed via the choice of the attention normalization and additional regularization.   
+5. On image, text and tabular data, we demonstrate the four key benefits of ProtoAttend: interpretability, confidence control, diagnosis of distribution mismatch, and robustness against label noise. ProtoAttend yields superior quality for sample-based interpretability, better-calibrated confidence scoring, and more sensitive out-of-distribution detection compared to alternative approaches.   
+6. ProtoAttend enables all these benefits via the same architecture and method, while maintaining comparable overall accuracy.
+
+# 2 RELATED WORK
+
+Prototypical learning: The principles of ProtoAttend are inspired by (Bien & Tibshirani, 2012). They formulate prototype selection as an integer program and solve it using a greedy approach with linear program relaxation. It seems unclear whether such approaches can be efficiently adopted to deep learning. (Chen et al., 2018) and (Li et al., 2018) introduce a prototype layer for interpretability by replacing the conventional inner product with a distance computation for perceptual similarity. In contrast, our method uses an attention mechanism to quantify perceptual similarity and can choose input-dependent prototypes from a large-scale candidate database. (Yeh et al., 2018) decomposes the prediction into a linear combination of activations of training points for interpretability using representer values. The linear decomposition idea also exists in ProtoAttend, but the weights are learned via an attention mechanism and sparsity is encouraged in the decomposition. In (Koh & Liang, 2017), the training points that are the most responsible for a given prediction are identified using influence functions via oracle access to gradients and Hessian-vector products.
+
+Metric learning: Metric learning aims to find an embedding representation of the data where similar data points are close and dissimilar data pointers are far from each other. ProtoAttend is motivated by efficient learning of such an embedding space which can be used to decompose decisions. Metric learning for deep neural networks is typically based on modifications to the objective function, such as using triplet loss and N-pair loss (Sohn, 2016; Cui et al., 2016; Hoffer & Ailon, 2014). These yield perceptually meaningful embedding spaces yet typically require a large subset of nearest neighbors to avoid degradation in performance (Cui et al., 2016). (Kim et al., 2018) proposes a deep metric learning framework which employs an attention-based ensemble with a divergence loss so that each learner can attend to different parts of the object. Our method has metric learning capabilities like relating similar data points, but also performs well on the ultimate supervised learning task.
+
+Attention-based few-shot learning: Some of our inspirations are based on recent advances in attention-based few-shot learning. In (Vinyals et al., 2016), an attention mechanism is used to relate an example with candidate examples from a support set using a weighted nearest-neighbor classifier applied within an embedding space. In (Ren et al., 2018), incremental few-shot learning is implemented using an attention attractor network on the encoded and support sets. In (Snell et al., 2017), a non-linear mapping is learned to determine the prototype of a class as the mean of its support set in the embedding space. During training, the support set is randomly sampled to mimic the inference task. Overall, the attention mechanism in our method follows related principles but fundamentally differs in that few-shot learning aims for generalization to unseen classes whereas the goal of our method is robust and interpretable learning for seen classes.
+
+Uncertainty and confidence estimation: ProtoAttend takes a novel perspective on the perennial problem of quantifying how much deep neural networks’ predictions can be trusted. Common approaches are based on using the scores from the prediction model, such as the probabilities from the softmax layer of a neural network, yet it has been shown that the raw confidence values are typically poorly calibrated (Guo et al., 2017). Ensemble of models (Lakshminarayanan et al., 2017) is one of the simplest and most efficient approaches, but significantly increases complexity and decreased interpretability. In (Papernot & McDaniel, 2018), the intermediate representations of the network are used to define a distance metric, and a confidence metric is proposed based on the conformity of the neighbors. (Jiang et al., 2018), proposes a confidence metric based on the agreement between the classifier and a modified nearest-neighbor classifier on the test sample. In (DeVries & Taylor, 2018), direct inference of confidence output is considered with a modified loss. Another direction of uncertainty and confidence estimation is Bayesian neural networks that return a distribution over the outputs (Kendall & Gal, 2017b) (Mullachery et al., 2018) (Kendall & Gal, 2017a).
+
+# 3 PROTOATTEND: ATTENTION-BASED PROTOTYPICAL LEARNING
+
+Consider a training set with samples, $\mathcal { T } = \{ \mathbf { x _ { i } } , y _ { i } \}$ . Conventional supervised learning aims to learn a model $s ( \mathbf { x _ { i } } ; \mathbf { S } )$ that minimizes a predefined loss $\begin{array} { r } { \dot { 1 } / B \cdot \sum _ { i = 1 } ^ { B } L ( y _ { i } , \hat { y _ { i } } = s ( \mathbf x _ { \mathbf i } ; \mathbf S ) ) ^ { 1 } } \end{array}$ at each iteration, where $B$ is the batch size for training. Our goal is to impose that decision making should be based on only a small number of training examples, i.e. prototypes, such that their linear superposition in an embedding space can yield the overall decision and the superposition weights correspond to their importance. Towards this goal, we propose defining a solution to prototypical learning with the following six principles:
+
+i. $\mathbf { v _ { i } } = f ( \mathbf { x _ { i } } ; \theta )$ encodes all relevant information of $\mathbf { x _ { i } }$ for the final decision. $f ( )$ considers the global distribution of the samples, i.e. learns from all $\{ \mathbf { x _ { i } } , y _ { i } \}$ . Although all the information in training dataset is embodied in the weights of the encoder2, we construct the learning method in such a way that decision is dominated by the prototypes with high weights. ii. From the encoded information, we can find a decision function so that the mapping $g ( \mathbf { v _ { i } } ; \eta )$ is close to the ground truth $y _ { i }$ , in a consistent way with conventional supervised learning.   
+iii. Given candidates $\mathbf { x _ { j } ^ { ( c ) } }$ to select the prototypes from, there exists weights $p _ { i , j }$ (where $p _ { i , j } \geq 0$ and is cl $\textstyle \sum _ { j = 1 } ^ { D } p _ { i , j } = 1 )$ , such td truth t the decision . $g ( \sum _ { j = 1 } ^ { D } p _ { i , j } \mathbf { v _ { j } ^ { ( c ) } } ; \eta )$ (where ${ \bf v _ { j } ^ { ( c ) } } = f ( { \bf x _ { j } ^ { ( c ) } } ; \theta ) )$ $y _ { i }$   
+iv. When the linear combination $\textstyle \sum _ { j = 1 } ^ { D } p _ { i , j } \mathbf { v _ { j } ^ { ( c ) } }$ is considered, prototypes with higher weights $p _ { i , j }$ have higher contribution in the decision $g ( \sum _ { j = 1 } ^ { D } p _ { i , j } \mathbf { v _ { j } ^ { ( c ) } } ; \eta )$ v. The weights should be sparse – only a controllable amount of weights $p _ { i , j }$ should be nonzero. Ideally, there exists an efficient mechanism for outputting $p _ { i , j }$ to control the sparsity without significantly affecting performance.   
+vi. The weights $p _ { i , j }$ depend on the relation between input and the candidate samples, $p _ { i , j } =$ $r ( \mathbf { x _ { i } } , \mathbf { x _ { j } ^ { ( c ) } } ; \mathbf { r } )$ , based on their perceptual relation for decision making. We do not introduce any heuristic relatedness metric such as distances in the representation space, but we allow the model to learn the relation function that helps the overall performance.
+
+Learning involves optimization of the parameters $\theta ,  { \mathbf { \Gamma } } , \eta$ of the corresponding functions. If the proposed principles (such as reasoning from the linear combination of embeddings or assigning relevance to the weights) are not imposed during training but only at inference, a high performance cannot be obtained due to the train-test mismatch, as the intermediate representations can be learned in an arbitrary way without any necessities to satisfy them.3 The subsequent section presents ProtoAttend and training procedure to implement it.
+
+# 3.1 NETWORK ARCHITECTURE AND TRAINING
+
+The principles above are conditioned on efficient learning of an encoding function to encode the relevant information for decision making, a relation function to determine the prototype weights, and a final decision making block to return the output. Conventional supervised learning comprises the encoding and decision blocks. On the other hand, it is challenging to design a learning method with a relation function with a reasonable complexity. To this end, we adapt the idea of attention (Corbetta & Shulman, 2002; Vaswani et al., 2017), where the model focuses on an adaptive small portion of input while making the decision. Different from conventional employment of attention in sequence or visual learning, we propose to use attention at sample level, such that the attention mechanism is used to determine the prototype weights by relating the input and the candidate samples via alignment of their keys and queries. Fig. 2 shows the proposed architecture for training and inference. The three main blocks are described below:
+
+![](images/c44b480160305586dc6788fbb6715a797e9d7cd241009c74c79f4614711ebfd6.jpg)  
+Figure 2: ProtoAttend method for training and testing. Shared encoder between input samples and the candidate samples generates input representations, that are mapped to key, query and value embeddings (with a single nonlinear layer). The alignment between keys and queries determines the weights of the prototypes, and the linear combination of the values determines the final decision. Conformity of the prototype labels is used as a confidence metric.
+
+Encoder: A trainable encoder is employed to transform $B$ input samples (note that $B$ may be 1 at inference) and $D$ samples from the database of prototype candidates (note that $D$ may be as large as the entire training dataset at inference) into keys, queries and values. The encoder is shared and jointly updated for the input samples and prototype candidate database, to learn a common representation space for the values. The encoder architecture can be based on any trainable discriminative feature mapping function, e.g. ResNet (He et al., 2016) for images, with the modification of generating three types of embeddings. For mapping of the last encoder layer to key, query and value embeddings, we simply use a single fully-connected layer with a nonlinearity, separately for each.4 For input samples, $\mathbf { \bar { V } } \in \mathfrak { R } ^ { B \times d _ { o u t } }$ and $\mathbf { \check { Q } } \in \Re ^ { B \times d _ { a t t } }$ denote the values and queries, and for candidate database samples ${ \bf K } ^ { ( { \bf c } ) } \in \Re ^ { D \times d _ { a t t } }$ and $\mathbf { V } ^ { ( \mathbf { c } ) } \in \mathfrak { R } ^ { D \times d _ { o u t } }$ denote the keys and values. For keys and queries, we use separate representations as the entire system is not symmetric, there are a lot of candidate samples and the model may prefer to learn the keys to arrange the representation space such that it is meaningful when their inner products with a single query are considered.
+
+Relational attention: The relational attention yields the weight between the $i ^ { t h }$ sample and $j ^ { t h }$ candidate, $p _ { i , j }$ , via alignment of the corresponding key and query in dot-product attention form5:
+
+$$
+p _ { i , j } = n \left( \mathbf { K _ { j } ^ { ( c ) } } \mathbf { Q _ { i } } ^ { T } / \sqrt { d _ { a t t } } \right) ,
+$$
+
+where $n ( )$ is a normalization function to satisfy $p _ { i , j } \geq 0$ and $\textstyle \sum _ { j = 1 } ^ { D } p _ { i , j } = 1$ for which we consider softmax and sparsemax (Martins & Astudillo, $2 0 1 6 ) ^ { 6 }$ . The choice of the normalization function is an efficient mechanism to control the sparsity of the prototype weights, as demonstrated in experiments. Note that the relational attention mechanism does not introduce any extra trainable parameters.
+
+Decision making: The final decision block simply consists of a linear mapping from a convex combination of values that results in the output $y _ { i }$ . Consider the convex combination of value embeddings, parameterized by $\alpha$ :
+
+$$
+\hat { y } _ { i } ( \alpha ) = g \left( ( 1 - \alpha ) \mathbf { v _ { i } } + \alpha \sum _ { j = 1 } ^ { D } p _ { i , j } \mathbf { v _ { j } ^ { ( c ) } } \right) .
+$$
+
+For $\alpha = 0$ , $L \left( y _ { i } , \hat { y _ { i } } ( 0 ) \right)$ is the conventional supervised learning loss (ignoring the relational attention mechanism) that can only impose principles (i) and (ii), but not the principles (iii)-(vi). A high accuracy for $\hat { y _ { i } } ( 0 )$ merely indicates that the value embedding space represents each input sample accurately. For $\alpha = 1$ , $\dot { L } \left( y _ { i } , \hat { y _ { i } } ( 1 ) \right)$ encourages the principles (i), (iii)-(iv), but not the principles (ii) and (vi).7 A high accuracy for $\hat { y _ { i } } ( 1 )$ indicates that the linear combination of value embeddings accurately maps to the decision. For (vi), we propose that there should be a similar output mapping for the input and prototypes, for which we encourage high accuracy for both $\hat { y } _ { i } ( 0 )$ and $\hat { y _ { i } } ( 1 )$ with a loss term that is a mixture of $L \left( y _ { i } , \hat { y _ { i } } ( 0 ) \right)$ and $L \left( y _ { i } , { \hat { y _ { i } } } ( 1 ) \right)$ or guidance with an intermediate term, as ${ \hat { y } } _ { i } ( 0 . 5 )$ , is required. Lastly, when $\alpha \leq 0 . 5$ , we obtain the condition that the input sample itself has the largest contribution in the linear combination. Intuitively, the sample itself should be more relevant for the output compared to other samples, so the principles (iii) and (iv) can be encouraged. We propose and compare different training objective functions in Table 1. We observe that the last four are all viable options as the training objective, with similar performance. We choose the last one for the rest of the experiments, as in some cases, slightly better prototypes are observed qualitatively (see Sect. 5.2 for further discussion).
+
+Table 1: Ablation study. Impact of various training losses on ProtoAttend with softmax attention for Fashion-MNIST. $1 \leq i \leq N _ { t }$ is the training iteration index and $N _ { t }$ is the total number of iterations.   
+
+<table><tr><td rowspan=1 colspan=1>Training objective function</td><td rowspan=1 colspan=1>Acc.%for yi(0)</td><td rowspan=1 colspan=1>Acc.%for yi(1)</td><td rowspan=1 colspan=1>-E{C}g=y</td><td rowspan=1 colspan=1>-E{C}gy</td></tr><tr><td rowspan=1 colspan=1>L (yi,yi(O))</td><td rowspan=1 colspan=1>94.28</td><td rowspan=1 colspan=1>13.13</td><td rowspan=1 colspan=1>0.029</td><td rowspan=1 colspan=1>0.194</td></tr><tr><td rowspan=1 colspan=1>L(yi，yi(1))</td><td rowspan=1 colspan=1>10.92</td><td rowspan=1 colspan=1>94.21</td><td rowspan=1 colspan=1>0.103</td><td rowspan=1 colspan=1>0.002</td></tr><tr><td rowspan=1 colspan=1>L (yi, yi(0.5))</td><td rowspan=1 colspan=1>94.01</td><td rowspan=1 colspan=1>94.25</td><td rowspan=1 colspan=1>0.927</td><td rowspan=1 colspan=1>0.049</td></tr><tr><td rowspan=1 colspan=1>L(yi,yi(O))+L(yi,yi(1))</td><td rowspan=1 colspan=1>94.37</td><td rowspan=1 colspan=1>94.38</td><td rowspan=1 colspan=1>0.931</td><td rowspan=1 colspan=1>0.047</td></tr><tr><td rowspan=1 colspan=1>(1-i/Nt)·L(yi,y(O)) +(i/Nt)·L(yi,yi(1))</td><td rowspan=1 colspan=1>94.14</td><td rowspan=1 colspan=1>94.18</td><td rowspan=1 colspan=1>0.927</td><td rowspan=1 colspan=1>0.049</td></tr><tr><td rowspan=1 colspan=1>L(yi,yi(O)) +L(yi，,yi(1)) +L(yi,yi(0.5))</td><td rowspan=1 colspan=1>94.37</td><td rowspan=1 colspan=1>94.45</td><td rowspan=1 colspan=1>0.928</td><td rowspan=1 colspan=1>0.047</td></tr></table>
+
+To control the sparsity of the weights (beyond the choice of the attention operation), we also propose a sparsity regularization term with a coefficient $\lambda _ { s p a r s e }$ in the form of entropy, $L _ { s p a r s e } ( \mathbf { p } ) =$ $- 1 / B \textstyle \sum _ { i = 1 } ^ { B } \sum _ { j = 1 } ^ { D } p _ { i , j } \log ( p _ { i , j } + \epsilon )$ , where $\epsilon$ is a small number for numerical stability. $L _ { s p a r s e } ( \mathbf { p } )$ is minimized when $\mathbf { p }$ has only 1 non-zero value.
+
+# 3.2 CONFIDENCE SCORING USING PROTOTYPES
+
+![](images/bce2b5a023f7fb5db212946b111d8d7baf40321a778282f3f65b4015a9d4ecdb.jpg)  
+Figure 3: Impact of confidence on ProtoAttend accuracy. Reliability diagram for Fashion-MNIST, as in (Papernot & McDaniel, 2018). Bars (left axis) indicate the mean accuracy of predictions binned by confidence; the red line (right axis) shows the number of samples across bins.
+
+ProtoAttend provides a linear decomposition (via value embeddings) of the decision into prototypes that have known labels. Ideally, labels of the prototypes should all be the same as the labels of the input. When prototypes with high weights belong to the same class, the model shall be more confident and a correct classification result is expected, whereas in the cases of disagreement between prototype labels, the model shall be less confident and the likelihood of a wrong prediction is higher. With the motivation of separating correct vs. incorrect decisions via its value, we propose a confidence score based on the agreement between the prototypes:
+
+$$
+C _ { i } = \sum _ { j = 1 } ^ { D } p _ { i , j } \cdot \operatorname { I } ( y _ { j } ^ { ( c ) } = \hat { y _ { i } } ) ,
+$$
+
+where $I ( )$ is the indicator function. Table 1 shows the significant difference of the average confidence metric between correct vs. incorrect classification cases for the test dataset, as desired. In Fig. 3, the impact of confidence on accuracy is further analyzed with the reliability diagram as in (Papernot & McDaniel, 2018). When test samples are binned according to their confidence, it is observed that the bins with higher confidence yield much higher accuracy. There are small number of samples in the bins with lower confidence, and those tend to be the incorrect classification cases. In Section 4.4, the efficacy of confidence score in separating correct vs. incorrect classification is experimented in confidence-controlled prediction setting, demonstrating how much the prediction accuracy can be improved by refraining from small number of samples with low confidence at test time.
+
+To further encourage confidence during training, we also consider a regularization term $- 1 / B \textstyle \sum _ { i = 1 } ^ { B } \sum _ { j = 1 } ^ { D } p _ { i , j } \cdot \operatorname { I } ( y _ { j } ^ { ( c ) } = y _ { i } )$ with a coefficient $\lambda _ { c o n f }$ . $L _ { c o n f }$ s minimized when all proto-8 $L _ { c o n f } ( \mathbf { p } ) =$ types with $p _ { i , j } > 0$ are from the same ground truth class with output $y _ { i }$
+
+# 4 EXPERIMENTS
+
+# 4.1 SETUP
+
+We demonstrate the results of ProtoAttend for image, text and tabular data classification problems with different encoder architectures (see Supplementary Material for details). Outputs of the encoders are mapped to queries, keys and values using a fully-connected layer followed by ReLU. For values, layer normalization (Lei Ba et al., 2016) is employed for more stable training. A fully-connected layer is used in the decision making block, yielding logits for determining the estimated class. Softmax cross entropy loss is used as $L ( )$ . Adam optimization algorithm is employed (Kingma & Ba, 2014) with exponential learning rate decay (with parameters optimized on a validation set). For image encoding, unless specified, we use the standard ResNet model (He et al., 2016). For text encoding, we use the very deep convolutional neural network (VDCNN) (Conneau et al., 2016) model, inputting sequence of raw characters. For tabular data encoding, we use an LSTM model (Hochreiter & Schmidhuber, 1997), which inputs the feature embeddings at every timestep. See Supplementary Material for implementation details, additional results and discussions.
+
+# 4.2 SPARSE EXPLANATIONS OF DECISIONS
+
+We foremost demonstrate that our inherently-interpretable model design does not cause significant degradation in performance. Table 2 shows the accuracy and the median number of prototypes required to add up to a particular portion of the decision9 for different prototypical learning cases. In all cases, very small accuracy gap is observed with the baseline encoder that is trained in conventional supervised learning way. The attention normalization function and sparsity regularization are efficient mechanisms to control the sparsity – the number of prototypes required is much lower with sparsemax attention compared to softmax attention and can be further reduced with sparsity regularization (see Supplementary Material for details). With a small decrease in performance, the number of prototypes can be reduced to just a handful.10 There is difference between datasets, as intuitively expected from the discrepancy in the degree of similarity between the intra-class samples.
+
+Table 2: ProtoAttend achieves interpretability without significant degradation in performance. Accuracy and median number of prototypes to add up to $50 \%$ , $90 \%$ and $9 5 \%$ of the decision, quantified with prototype weights.   
+
+<table><tr><td rowspan=2 colspan=1>Dataset</td><td rowspan=2 colspan=1>Method</td><td rowspan=2 colspan=1>Acc.%</td><td rowspan=1 colspan=3>No. of prototypes</td></tr><tr><td rowspan=1 colspan=3>50 %  90 %  95%</td></tr><tr><td rowspan=3 colspan=1>MNIST</td><td rowspan=1 colspan=1>Baseline enc.</td><td rowspan=1 colspan=1>99.70</td><td rowspan=1 colspan=3>1</td></tr><tr><td rowspan=1 colspan=1>Softmax attn.</td><td rowspan=1 colspan=1>99.66</td><td rowspan=1 colspan=1>365</td><td rowspan=1 colspan=1>1324</td><td rowspan=1 colspan=1>1648</td></tr><tr><td rowspan=1 colspan=1>Sparsemax attn.</td><td rowspan=1 colspan=1>99.69</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=4 colspan=1>Fashion-MNIST</td><td rowspan=1 colspan=1>Baseline enc.</td><td rowspan=1 colspan=1>94.74</td><td rowspan=1 colspan=3>1</td></tr><tr><td rowspan=1 colspan=1>Softmax attn.</td><td rowspan=1 colspan=1>94.42</td><td rowspan=1 colspan=1>712</td><td rowspan=1 colspan=1>2320</td><td rowspan=1 colspan=1>2702</td></tr><tr><td rowspan=1 colspan=1>Sparsemax attn.</td><td rowspan=1 colspan=1>94.42</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>11</td></tr><tr><td rowspan=1 colspan=1>Sparsemax attn. + sparsity reg.</td><td rowspan=1 colspan=1>94.47</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=4 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>Baseline enc.</td><td rowspan=1 colspan=1>91.97</td><td rowspan=1 colspan=3>-</td></tr><tr><td rowspan=1 colspan=1>Softmax attn.</td><td rowspan=1 colspan=1>91.69</td><td rowspan=1 colspan=1>317</td><td rowspan=1 colspan=1>1453</td><td rowspan=1 colspan=1>1898</td></tr><tr><td rowspan=1 colspan=1>Sparsemax attn.</td><td rowspan=1 colspan=1>91.44</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>14</td><td rowspan=1 colspan=1>16</td></tr><tr><td rowspan=1 colspan=1>Sparsemax attn. + sparsity reg.</td><td rowspan=1 colspan=1>91.26</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td></tr><tr><td rowspan=3 colspan=1>DBPedia</td><td rowspan=1 colspan=1>Baseline enc.</td><td rowspan=1 colspan=1>98.25</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Softmax attn.</td><td rowspan=1 colspan=1>98.20</td><td rowspan=1 colspan=1>63</td><td rowspan=1 colspan=1>190</td><td rowspan=1 colspan=1>225</td></tr><tr><td rowspan=1 colspan=1>Sparsemax attn.</td><td rowspan=1 colspan=1>97.74</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td></tr><tr><td rowspan=4 colspan=1>Income</td><td rowspan=1 colspan=1>Baseline enc.</td><td rowspan=1 colspan=1>85.68</td><td rowspan=1 colspan=3>-</td></tr><tr><td rowspan=1 colspan=1>Softmax attn.</td><td rowspan=1 colspan=1>85.64</td><td rowspan=1 colspan=1>2263</td><td rowspan=1 colspan=1>9610</td><td rowspan=1 colspan=1>12419</td></tr><tr><td rowspan=1 colspan=1>Sparsemax attn.</td><td rowspan=1 colspan=1>85.58</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>57</td><td rowspan=1 colspan=1>67</td></tr><tr><td rowspan=1 colspan=1>Sparsemax attn. + sparsity reg.</td><td rowspan=1 colspan=1>85.41</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>7</td></tr></table>
+
+![](images/6067e1c598a850d0865f30ad7d8c4e0fe47cff0b6d4c9585b1c9d89d33c59be8.jpg)  
+Figure 4: Example inputs and ProtoAttend prototypes for (a) MNIST (with sparsemax), FashionMNIST dataset (with sparsemax and sparsity regularization) and (b) Fruits (with sparsemax and sparsity regularization). For MNIST & Fashion-MNIST, prototypes typically consist of discriminative features such as the straight line shape for the digit 1, and the long heels and strips for the sandal. For Fruits, prototypes often correspond to the same fruit captured from a very similar angle.
+
+![](images/b77e2ea292c994b192a4a05347b446e6a013a7768cee836590131973ae5bae53.jpg)  
+Figure 5: Example inputs and ProtoAttend prototypes for DBPedia (with sparsemax). While classifying the inputs as athlete, prototypes have very similar sentence structure, words and concepts.
+
+# Inputs
+
+# Prototypes
+
+![](images/8ce470d06f2745621153b260151b1e2105ba58828a88749ef817fcec823f5a20.jpg)
+
+Figure 6: Example inputs and ProtoAttend prototypes for Adult Census Income (with sparsemax and sparsity regularization). For the first example, all prototypes have similar age, two share similar education level and one has the same occupation. For the second example, three prototypes have the same occupation, all work more than 40 hours/week, and three have postgraduate education.
+
+Figs. 4, 5 and 6 exemplify prototypes for image, text and tabular data. In general, perceptually-similar samples are chosen as the prototypes with the largest weights. We also compare the relevant samples found by ProtoAttend with the methods of representer point selection (Yeh et al., 2018) and influence functions (Koh & Liang, 2017) (see Supplementary Material for details) on Animals with Attributes dataset. As shown in Fig. 7, our method finds qualitatively more relevant samples. This case also exemplifies the potential of our method for integration into pre-trained models by addition of simple layers for key, query and value generation.
+
+![](images/10d0a87a1ec524026d704164a749e6f06d6a2dd6b8d5967a139f41d9643803e7.jpg)  
+Figure 7: Samples found by ProtoAttend vs. representer point selection (Yeh et al., 2018) and influence function (Koh & Liang, 2017) for the two examples from (Yeh et al., 2018) on Animals with Attributes dataset. See Supplementary Material for more examples.
+
+4.3 ROBUSTNESS TO LABEL NOISE
+
+Table 3: Label noise ratio vs. accuracy for baseline encoder, dropout method (Arpit et al., 2017) (optimizing the keep probability) and ProtoAttend with sparsemax attention and sparsity regularization for CIFAR-10.   
+
+<table><tr><td rowspan=2 colspan=1>Noise level</td><td rowspan=1 colspan=3>Test accuracy%</td></tr><tr><td rowspan=1 colspan=1>Baseline</td><td rowspan=1 colspan=1>Dropout</td><td rowspan=1 colspan=1>ProtoAttend</td></tr><tr><td rowspan=1 colspan=1>0.8</td><td rowspan=1 colspan=1>57.02</td><td rowspan=1 colspan=1>56.76</td><td rowspan=1 colspan=1>60.50</td></tr><tr><td rowspan=1 colspan=1>0.6</td><td rowspan=1 colspan=1>71.27</td><td rowspan=1 colspan=1>72.15</td><td rowspan=1 colspan=1>74.67</td></tr><tr><td rowspan=1 colspan=1>0.4</td><td rowspan=1 colspan=1>77.47</td><td rowspan=1 colspan=1>78.99</td><td rowspan=1 colspan=1>80.04</td></tr></table>
+
+As prototypical learning with sparsemax attention aims to extract decision-making information from a small subset of training samples, it can be used to improve performance when the training dataset contains noisy labels (see Table 3). The optimal value11 of $\lambda _ { s p a r s e }$ increases with higher noisy label ratios, underlining the increasing importance of sparse learning.
+
+# 4.4 CONFIDENCE-CONTROLLED PREDICTION
+
+![](images/63d10481d6635174332b841e716f6182533446db42924cf1efe71cf2f21185ab.jpg)  
+Figure 8: Confidence-controlled prediction. (a) Accuracy vs. ratio of samples for MNIST. We compare dkNN (Papernot & McDaniel, 2018) and prototypical learning (with softmax attention and $\lambda _ { c o n f } { = } 0 . 1 $ using the same network architecture from (Papernot & McDaniel, 2018) without augmentation. (b) Accuracy vs. ratio of samples for CIFAR-10. We compare prototypical learning (with softmax attention and $\lambda _ { c o n f } { = } 0 . 1 )$ with trust score (Jiang et al., 2018) and deep ensemble (Lakshminarayanan et al., 2017) methods for the same baseline encoder network architecture.
+
+By varying the threshold for the confidence metric, a trade-off can be obtained for what ratio of the test samples that the model makes a prediction for vs. the overall accuracy it obtains on the samples above that threshold.12 Figs. 8(a) and 8(b) demonstrate this trade-off and compare it to alternative methods. The sharper slope of the plots show that our method is superior to dkNN (Papernot & McDaniel, 2018) and trust score (Jiang et al., 2018), the methods based on quantifying the mismatch with nearest-neighbor samples, in terms of finding related samples. Although the baseline accuracy is higher with 4 ensemble networks obtained via deep ensemble (Lakshminarayanan et al., 2017), our method utilizes a single network and the additional accuracy gains by refraining from uncertain predictions is similar to our approach as shown by the similar slopes of the curves.
+
+Overall, the baseline accuracy can be significantly improved by making less predictions. Compared to the state of the art models, our canonical method with simple and small models shows similar accuracy by making slightly fewer predictions – e.g. for MNIST, (Wan et al., 2013) achieves $0 . 2 1 \%$ error rate, that is obtained by our method refraining from only $0 . 4 5 \%$ of predictions using ResNet-32 and for DBpedia, (Sachan & Petuum, 2018) achieves $0 . 9 1 \%$ error, that is obtained by our method refraining from $3 \%$ of predictions using 9-layer VDCNN. In general, the smaller the number of prototypes, the smaller the trade-off space. Thus, softmax attention (which normally results in more prototypes) is better suited for confidence-controlled prediction compared to sparsemax (see Supplementary Material for more comparisons).
+
+# 4.5 OUT-OF-DISTRIBUTION SAMPLES
+
+Well-calibrated confidence scores at inference can be used to detect deviations from the training dataset. As the test distribution deviates from the training distribution, prototype weights tend to mismatch more and yield lower confidence scores. Fig. 9 (a) shows the ratio of samples above a certain confidence level as the test dataset deviates. Rotations deviate the distribution of test images from the training images, and cause significant degradation in confidence scores, as well as the overall accuracy. On the other hand, using test image from a different dataset, degrade them even further. Next, Fig. 9 (b) shows quantification of out-of-distribution detection with prototypical learning, using the method from (Hendrycks & Gimpel, 2016). ProtoAttend yields an AUC of 0.838, being on par with the-state of the art approaches (Hendrycks et al.).
+
+![](images/2c553620572a6e45f5fc0ab60b86dfead180af1d3ae2d36892ce2aed4decf6de.jpg)  
+Figure 9: Out-of-distribution detection. (a) Ratio of samples above the confidence level for prototypical learning with softmax attention, trained with Fashion-MNIST, and tested on the shown datasets. E.g. if we assess the ratio of samples above confidence 0.9, it is far more likely that those samples come from the same distribution with the training dataset. (b) ROC curve for in-distribution vs. out-of-distribution detection, using CIFAR-10 as in-distribution and SVHN as out-of-distribution, computed using the method from (Hendrycks & Gimpel, 2016) and compared to the proposed baseline in (Hendrycks & Gimpel, 2016). Softmax attention and confidence regularization $( \lambda _ { c o n f } = 0 . 1 )$ ) are used.
+
+# 5 COMPUTATIONAL COST
+
+ProtoAttend requires only a very small increase in the number of learning parameters (merely two extra small matrices for the fully-connected layers to obtain queries and keys). However, it does require a longer training time and has higher memory requirements to process the candidate database. At inference, keys and values for the candidate database can be computed only once and integrated into the model. Thus, the overhead merely becomes the computation of attention outputs (e.g. for CIFAR-10 model, the attention overhead at inference is less than 0.6 MFLOPs, orders of magnitude lower than the computational complexity of a ResNet model). During training on the other hand, both forward and backward propagation steps for the encoder need to be computed for all candidate samples and the total time is higher (e.g. 4.45 times slower to train until convergence for CIFAR-10 compared to the conventional supervised learning). The size of the candidate database is limited by the memory of the processor, so in practice we sample different candidate databases randomly from the training dataset at each iteration. For faster training, data and model parallelism approaches are straightforward to implement – e.g., different processors can focus on different samples, or they can focus on different parts of the convolution or inner product operations. Further computationally-efficient approaches may involve less frequent updates for candidate queries and values.
+
+# 6 CONCLUSIONS
+
+We propose an attention-based prototypical learning method, ProtoAttend, and demonstrate its usefulness for a wide range of problems on image, text and tabular data. By adding a relational attention mechanism to an encoder, prototypical learning enables novel capabilities. With sparsemax attention, it can base the learning on a few relevant samples that can be returned at inference for interpretability, and can also improve robustness to label noise. With softmax attention, it enables confidence-controlled prediction that can outperform state of the art results with simple architectures by simply making slightly fewer predictions, as well as enables detecting deviations from the training data. All these capabilities are achieved without sacrificing overall accuracy of the base model.
+
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+A PSEUDO CODE FOR TRAINING
+
+# Algorithm 1 Pseudo-code of ProtoAttend training
+
+1: Inputs: Training dataset $\tau$ , encoder model $h ( \mathbf { x } ; \theta )$ , classifier model $h ( \mathbf { v } ; \phi )$ , normalization function $n$ , input batch size $B$ , candidate batch size $D$ , attention dimension $d _ { a t t }$ , $\alpha$ values to be used for loss: (0, 0.5, 1), task-specific loss function $L$ , ADAM learning rate $r$ , and exponential decay rate parameters $\beta _ { 1 }$ and $\beta _ { 2 }$
+
+$\theta$ $\phi$
+
+3: while until convergence do
+
+4: Sample a mini-batch from the training dataset for the inputs: $( \mathbf { x } _ { i } , y _ { i } ) _ { i = 1 } ^ { B } \sim T$
+
+5: Sample a mini-batch from the training dataset for the prototypes: $( \mathbf { x _ { j } ^ { ( c ) } } , y _ { j } ^ { ( c ) } ) _ { j = 1 } ^ { D } \sim \mathcal { T }$   
+6: for $i = 1 , . . . , B$ do
+
+7: Obtain queries and values for the input:
+
+$$
+\mathbf { Q _ { i } } , \mathbf { V _ { i } }  h ( \mathbf { x } ; \theta )
+$$
+
+8: for $j = 1 , . . . , D$ do
+
+9: Obtain keys and values for the prototypes:
+
+$$
+\mathbf { K _ { j } ^ { ( c ) } } , \mathbf { V _ { j } ^ { ( c ) } } \gets h ( \mathbf { x } ^ { ( \mathbf { c } ) } ; \theta )
+$$
+
+10: for $i = 1 , . . . , B$ do   
+11: for $j = 1 , . . . , D$ do   
+12: Estimate the relational attention coefficients:
+
+$$
+p _ { i , j } \gets n \left( \mathbf { K _ { j } ^ { ( c ) } } \mathbf { Q _ { i } } ^ { T } / \sqrt { d _ { a t t } } \right)
+$$
+
+13: Obtain the predictions
+
+14: for $i = 1 , . . . , B$ do
+
+$$
+\hat { y } _ { i } ( \alpha = 0 ) \gets g \left( \mathbf { v _ { i } } ; \phi \right)
+$$
+
+$$
+{ \hat { y } } _ { i } ( \alpha = 0 . 5 ) \gets g \left( 0 . 5 { \bf v _ { i } } + 0 . 5 \sum _ { j = 1 } ^ { D } p _ { i , j } { \bf v _ { j } ^ { ( c ) } } ; \phi \right)
+$$
+
+$$
+\hat { y } _ { i } ( \alpha = 1 ) \gets g \left( \sum _ { j = 1 } ^ { D } p _ { i , j } \mathbf { v _ { j } ^ { ( c ) } } ; \boldsymbol { \phi } \right)
+$$
+
+15: Estimate the total loss function
+
+$$
+L _ { b a t c h } \gets 1 / B \cdot \sum _ { i = 1 } ^ { B } L \left( y _ { i } , \hat { y _ { i } } ( 0 ) \right) + L \left( y _ { i } , \hat { y _ { i } } ( 1 ) \right) + L \left( y _ { i } , \hat { y _ { i } } ( 0 . 5 ) \right)
+$$
+
+16: Update the encoder model and the classifier layer
+
+$$
+\begin{array} { r l } & { \phi  \phi - \mathrm { A D A M } ( \nabla _ { \phi } L _ { b a t c h } , r , \beta _ { 1 } , \beta _ { 2 } ) } \\ & { } \\ & { \theta  \theta - \mathrm { A D A M } ( \nabla _ { \theta } L _ { b a t c h } , r , \beta _ { 1 } , 2 ) } \end{array}
+$$
+
+# B RELATION TO INFLUENCE FUNCTIONS
+
+Here we clarify the relationship of our work to influence functions from a theoretical perspective. Influence functions quantify how a model’s predictions would change if we did not have a particular training point. For the purpose of sample-based explainability, (Koh & Liang, 2017) proposes that the relation between an input sample $\mathbf { x _ { i } }$ and the candidate samples13 $\bf { x } _ { j } ^ { ( c ) }$ can be obtained by quantifying the influence of upweighting $( \mathbf { x _ { j } ^ { ( c ) } } , y _ { j } ^ { ( c ) } )$ on the loss at a query point $( \mathbf { x _ { i } } , y _ { i } )$
+
+$$
+\mathcal { T } _ { i , j } = - \nabla _ { ( \theta , \phi ) } L ( y _ { j } ^ { ( c ) } ) ^ { T } ( \mathbf H _ { ( \hat { \theta } , \hat { \phi } ) } ^ { - 1 } ) ^ { T } \nabla _ { ( \theta , \phi ) } L ( y _ { i } ) ,
+$$
+
+where $\mathbf { H } _ { ( \hat { \theta } , \hat { \phi } ) }$ is the Hessian and is positive definite by assumption. Let’s consider the singular value decomposition $( \mathbf { H } _ { ( \hat { \theta } , \hat { \phi } ) } ^ { - 1 } ) = \Xi \cdot \Sigma \cdot \Psi ^ { T }$ and also define the function $k ( \mathbf { x } , y ) = \nabla _ { ( \theta , \phi ) } L ( y )$ . Then, Eq. 4 can be written as:
+
+$$
+\mathcal { T } _ { i , j } = ( \boldsymbol { \Psi } ^ { T } \cdot k ( \mathbf { x _ { j } ^ { ( c ) } } , y _ { j } ^ { ( c ) } ) ) ^ { T } \cdot ( - \Sigma \cdot \Xi ^ { T } ) \cdot k ( \mathbf { x _ { i } } , y _ { i } ) ,
+$$
+
+We can observe that $\mathcal { T } _ { i , j }$ is in the form of an inner product between two functions applied on $\left( \mathbf { x _ { i } } , y _ { i } \right)$ and $( \mathbf { x _ { j } ^ { ( c ) } } , y _ { j } ^ { ( c ) } )$ y(c)j ). These two functions are composed of a shared (and potentially complex) function, followed by a linear mapping with non-shared parameters. This expression is indeed in a similar form with the argument of the normalization function for attention in Eq. 1, where the queries and keys are obtained by a shared encoder except the last layer. The only notable difference is that ProtoAttend encoder functions merely input xi and x(c)j , not the ground truth labels. Instead of relying on ground truth labels or complex Hessian estimations, ProtoAttend infers the encoded representations for the queries and keys directly in a feedforward way, by learning from the entire training dataset. Note that ProtoAttend does not use a separate encoder for values, and obtains a high performance by sharing the vast majority of the parameters while obtaining the keys, queries and values.
+
+In (Koh & Liang, 2017), Influence Functions are also related to nearest neighbor search-based relevant point determination approaches, for sample-based explainability. When Euclidean space is considered for distances, with the assumption that all points have the same norm, the inner product between the representations correspond to their similarity. This scenario is the special case of ProtoAttend when we use the same representation for keys, queries and values, and when we train with only $\alpha = 0$ loss term although we would use $p _ { i , j }$ for similarity determination. As studied in (Koh & Liang, 2017), nearest neighbor-based methods are far less accurate in capturing the effect of model training, compared to Influence Functions. Our empirical results in Figs. 7 and 13 show superior performance of ProtoAttend compared to Influence Functions in finding perceptually more similar samples.
+
+Overall, unlike Influence Functions, ProtoAttend modifies the model training for the desired goals, that fundamentally yields more degrees of freedom to optimize while achieving superior prototype learning quality effectively.
+
+# C TRAINING DETAILS
+
+Different candidate databases are sampled randomly from the training dataset at each iteration. Training database size is chosen to fit the model to the memory of a single GPU. $D$ at inference is chosen sufficiently large to obtain high accuracy. Table 4 shows the database size $D$ for the datasets used in the experiments. The size of the prototype candidate database should be sufficiently large such that the model can attend to reasonable prototypes with high coefficients (separately for each input). With appropriate sparsity mechanisms, we normally only end up with a few prototypes with large coefficients. Indeed, most of the coefficients would be zero with sparsemax activation and sparsity regularization.
+
+Table 4: Datasets and database size D.   
+
+<table><tr><td rowspan=2 colspan=1>Dataset</td><td rowspan=2 colspan=1>Encoder</td><td rowspan=1 colspan=2>Database size D</td></tr><tr><td rowspan=1 colspan=1>Training</td><td rowspan=1 colspan=1>Inference</td></tr><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>ResNet</td><td rowspan=1 colspan=1>1024</td><td rowspan=1 colspan=1>32768</td></tr><tr><td rowspan=1 colspan=1>Fashion-MNIST</td><td rowspan=1 colspan=1>ResNet</td><td rowspan=1 colspan=1>1024</td><td rowspan=1 colspan=1>32768</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>ResNet</td><td rowspan=1 colspan=1>1024</td><td rowspan=1 colspan=1>32768</td></tr><tr><td rowspan=1 colspan=1>Fruits</td><td rowspan=1 colspan=1>ResNet</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>4096</td></tr><tr><td rowspan=1 colspan=1>ISICMelanoma</td><td rowspan=1 colspan=1>ResNet</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>4096</td></tr><tr><td rowspan=1 colspan=1>DBPedia</td><td rowspan=1 colspan=1>VDCNN</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>4096</td></tr><tr><td rowspan=1 colspan=1>Census Income</td><td rowspan=1 colspan=1>LSTM</td><td rowspan=1 colspan=1>4096</td><td rowspan=1 colspan=1>15360</td></tr></table>
+
+# C.1 IMAGE DATA
+
+# C.1.1 MNIST DATASET
+
+We apply random cropping after padding each side by 2 pixels and per image standardization. The base encoder uses a standard 32 layer ResNet architecture. The number of filters is initially 16 and doubled every 5 blocks. In each block, two $3 \times 3$ convolutional layers are used to transform the input, and the transformed output is added to the input after a $1 \times 1$ convolution. $4 \times$ downsampling is applied by choosing the stride as 2 after $5 ^ { t h }$ and $1 0 ^ { t h }$ blocks. Each convolution is followed by batch normalization and ReLU nonlinearity. After the last convolution, $7 \times 7$ average pooling is applied. The output is followed by a fully-connected layer of 256 units and ReLU nonlinearity, followed by layer normalization (Lei Ba et al., 2016). Keys and queries are mapped from the output using a fully-connected layer followed by ReLU nonlinearity, where the attention size is $d _ { a t t } { = } 1 6$ . Values are mapped from the output using a fully-connected layer of $d _ { o u t } { = } 6 4$ units and ReLU nonlinearity, followed by layer normalization. For the baseline encoder, the initial learning rate is chosen as 0.002 and exponential decay is applied with a rate of 0.9 applied every $^ \mathrm { 6 k }$ iterations. The model is trained for 84k iterations. For prototypical learning model with softmax attention, the initial learning rate is chosen as 0.002 and exponential decay is applied with a rate of 0.8 applied every 8k iterations. The model is trained for 228k iterations. For prototypical learning model with sparsemax attention, the initial learning rate is chosen as 0.001 and exponential decay is applied with a rate of 0.93 applied every 6k iterations. The model is trained for $2 2 8 \mathrm { k }$ iterations. All models use a batch size of 128 and gradient clipping above 20.
+
+# C.1.2 FASHION-MNIST DATASET
+
+We apply random cropping after padding each side by 2 pixels, random horizontal flipping, and per image standardization. The base encoder uses a standard 32 layer ResNet architecture, similar to our MNIST experiments. For the baseline encoder, the initial learning rate is chosen as 0.0015 and exponential decay is applied with a rate of 0.9 applied every $1 0 \mathrm { k }$ iterations. The model is trained for 332k iterations. For prototypical learning with softmax attention, the initial learning rate is chosen as 0.0007 and exponential decay is applied with a rate of 0.92 applied every $^ { 8 \mathrm { k } }$ iterations. The model is trained for $4 5 0 \mathrm { k }$ iterations. For prototypical learning with sparsemax attention, the initial learning rate is chosen as 0.001 and exponential decay is applied with a rate of 0.9 applied every $^ { 8 \mathrm { k } }$ iterations. The model is trained for 392k iterations. For prototypical learning with sparsemax attention and sparsity regularization (with $\lambda _ { s p a r s e } = 0 . 0 0 0 3 )$ , the initial learning rate is chosen as 0.001 and exponential decay is applied with a rate of 0.94 applied every 8k iterations. $\lambda _ { c o n f } = 0 . 1$ is chosen when confidence regularization is applied. The model is trained for $4 4 0 \mathrm { k }$ iterations. All models use a batch size of 128 and gradient clipping above 20.
+
+# C.1.3 CIFAR-10 DATASET
+
+We apply random cropping after padding each side by 3 pixels, random horizontal flipping, random vertical flipping and per image standardization. The base encoder uses a standard 50 layer ResNet architecture. The number of filters is initially 16 and doubled every 8 blocks. In each block, two $3 \times 3$ convolutional layers are used to transform the input, and the transformed output is added to the input after a $1 \times 1$ convolution. $4 \times$ downsampling is applied by choosing the stride as 2 after $8 ^ { t h }$ and $1 \dot { 6 } ^ { t h }$ blocks. Each convolution is followed by batch normalization and the ReLU nonlinearity. After the last convolution, $8 \times 8$ average pooling is applied. The output is followed by a fully-connected layer of 256 units and the ReLU nonlinearity, followed by layer normalization (Lei Ba et al., 2016). Keys and queries are mapped from the output using a fully-connected layer followed by the ReLU nonlinearity, where the attention size is $d _ { a t t } { = } 1 6$ . Values are mapped from the output using a fully-connected layer of $d _ { o u t } { = } 1 2 8$ units and the ReLU nonlinearity, followed by layer normalization. For the baseline encoder, the initial learning rate is chosen as 0.002 and exponential decay is applied with a rate of 0.95 applied every 10k iterations. The model is trained for $9 4 0 \mathrm { k }$ iterations. For prototypical learning with softmax attention, the initial learning rate is chosen as 0.0035 and exponential decay is applied with a rate of 0.95 applied every 10k iterations. The model is trained for $6 2 5 \mathrm { k }$ iterations. For prototypical learning with sparsemax attention, the initial learning rate is chosen as 0.0015 and exponential decay is applied with a rate of 0.95 applied every 10k iterations. The model is trained for $9 0 5 \mathrm { k }$ iterations. For prototypical learning with sparsemax attention and sparsity regularization (with $\lambda _ { s p a r s e } = 0 . 0 0 0 0 8 )$ , the initial learning rate is chosen as 0.0015 and exponential decay is applied with a rate of 0.95 applied every $1 2 \mathrm { k }$ iterations. $\lambda _ { c o n f } = 0 . 1$ is chosen when confidence regularization is applied. The model is trained for $4 5 0 \mathrm { k }$ iterations. All models use a batch size of 128 and gradient clipping above 20.
+
+CIFAR-10 experiments with noisy labels. For CIFAR-10 experiments with noisy labels for the base encoder we only optimize the learning parameters. Noisy labels are sampled uniformly from the set of labels excluding the correct one. The baseline model with noisy label ratio of 0.8 uses an initial learning rate of 0.001, decayed with a rate of 0.92 every $^ \mathrm { 6 k }$ iterations, and is trained for $1 5 \mathrm { k }$ iterations. For the dropout approach, dropout with a rate of 0.1 is applied, and the model uses an initial learning rate of 0.002, decayed with a rate of 0.85 every $^ { 8 \mathrm { k } }$ iterations, and is trained for $2 4 \mathrm { k }$ iterations. The baseline model with noisy label ratio of 0.6 uses an initial learning rate of 0.002, decayed with a rate of 0.92 every $^ \mathrm { 6 k }$ iterations, and is trained for 12k iterations. For the dropout approach, dropout with a rate of 0.3 is applied, and the model uses an initial learning rate of 0.002, decayed with a rate of 0.92 every $^ \mathrm { 8 k }$ iterations, and is trained for $1 8 \mathrm { k }$ iterations. The baseline model with noisy label ratio of 0.4 uses an initial learning rate of 0.002, decayed with a rate of 0.92 every 6k iterations, and is trained for $1 5 \mathrm { k }$ iterations. For the dropout approach, dropout with a rate of 0.5 is applied, and the model uses an initial learning rate of 0.002, decayed with a rate of 0.92 every $^ \mathrm { 6 k }$ iterations, and is trained for $1 8 \mathrm { k }$ iterations. For experiments for the prototypical learning model with sparsemax attention, we optimize the learning parameters and $\lambda _ { s p a r s e }$ . For the model with noisy label ratio of 0.8, $\lambda _ { s p a r s e } = 0 . 0 0 1 5$ , initial learning rate is chosen as 0.0006 and exponential decay is applied with a rate of 0.95 applied every $^ \mathrm { 8 k }$ iterations. The model is trained for $1 0 8 \mathrm { k }$ iterations. For the model with noisy label ratio of 0.6, $\lambda _ { s p a r s e } = 0 . 0 0 0 5$ , initial learning rate is chosen as 0.001 and exponential decay is applied with a rate of 0.9 applied every $^ \mathrm { 8 k }$ iterations. The model is trained for $9 2 \mathrm { k }$ iterations. For the model with noisy label ratio of 0.4, $\lambda _ { s p a r s e } = 0 . 0 0 0 3$ , initial learning rate is chosen as 0.001 and exponential decay is applied with a rate of 0.9 applied every $^ \mathrm { 6 k }$ iterations. The model is trained for $1 2 2 \mathrm { k }$ iterations.
+
+# C.1.4 FRUITS DATASET
+
+We apply random cropping after padding each side by 5 pixels, random horizontal flipping, random vertical flipping and per image standardization. In the encoder, first, a downsampling with a convolutional layer is applied with a stride of 2, and using 16 filters, followed by a downsampling with max-pooling with a stride of 2. After obtaining the $2 5 \times 2 5$ inputs, a standard 32 layer ResNet architecture (similar to MNIST) is used, followed by a fully-connected layer of 128 units and the ReLU nonlinearity, followed by layer normalization (Lei Ba et al., 2016). Keys and queries are mapped from the output using a fully-connected layer followed by the ReLU nonlinearity, where the attention size is $d _ { a t t } { = } 1 6$ . Values are mapped from the output using a fully-connected layer of $d _ { o u t } { = } 6 4$ units and the ReLU nonlinearity, followed by layer normalization. W eight decay with a factor of 0.0001 is applied for the convolutional filters. The model uses a batch size of 128 and gradient clipping above 20.
+
+# C.1.5 ISIC MELANOMA DATASET
+
+The ISIC Melanoma dataset is formed from the ISIC Archive (ISIC, 2016) that contains over $1 3 \mathrm { k }$ dermoscopic images collected from leading clinical centers internationally and acquired from a variety of devices within each center. The dataset consists of skin images with labels denoting whether they contain melanoma or are benign. We construct the training and validation dataset using 15122 images (13511 benign and 1611 melanoma cases), and the evaluation dataset using 3203 images (2867 benign and 336 melanoma). While training, benign cases are undersampled in each batch to have 0.6 ratio including candidate database sets at training and inference. All images are resized to $1 2 8 \times 1 2 8$ pixels. We apply random cropping after padding each side by 8 pixels, random horizontal flipping, random vertical flipping and per image standardization. In the encoder, first, a downsampling with a convolutional layer is applied with a stride of 2, and using 16 filters, followed by a downsampling with max-pooling with a stride of 2. After obtaining the $3 2 \times 3 2$ inputs, the base encoder uses a standard 50 layer ResNet architecture (similar to CIFAR10), followed by a fully-connected layer of 128 units and the ReLU nonlinearity, followed by layer normalization (Lei Ba et al., 2016). Keys and queries are mapped from the output using a fully-connected layer followed by the ReLU nonlinearity, where the attention size is $d _ { a t t } { = } 1 6$ . Values are mapped from the output using a fully-connected layer of $d _ { o u t } { = } 6 4$ units and the ReLU nonlinearity, followed by layer normalization. For the baseline encoder, the initial learning rate is chosen as 0.002 and exponential decay is applied with a rate of 0.9 applied every 3k iterations. The model is trained for 220k iterations. For prototypical learning with softmax attention, the initial learning rate is chosen as 0.0006 and exponential decay is applied with a rate of 0.9 applied every $3 \mathrm { k }$ iterations. The model is trained for $1 4 7 \mathrm { k }$ iterations. For prototypical learning with sparsemax attention, the initial learning rate is chosen as 0.0006 and exponential decay is applied with a rate of 0.9 applied every 4k iterations. The model is trained for 166k iterations. All models use a batch size of 128 and gradient clipping above 20.
+
+# C.1.6 ANIMALS WITH ATTRIBUTES DATASET
+
+We train ProtoAttend with sparsemax attention using the features from a pre-trained ResNet-50 as provided in (Yeh et al., 2018). To map the pre-trained features, we simply insert a single fullyconnected layer with 256 units with ReLU nonlinearity and layer normalization, followed by the individual fully-connected layers of keys, queries and values (16, 16 and 64 units respectively with ReLU nonlinearity). Sparsity regularization is applied with $\lambda _ { s p a r s e } = 0 . 0 0 0 0 0 1$ . We train the model for $7 0 \mathrm { k }$ iterations. The initial learning rate is chosen as 0.0006 and exponential decay is applied with a rate of 0.8 applied every 10k iterations. A classification accuracy above $91 \%$ is obtained for the test set.
+
+# C.2 TEXT DATA
+
+# C.2.1 DBPEDIA DATASET
+
+There are 14 output classes: Company, Educational Institution, Artist, Athlete, Office Holder, Mean Of Transportation, Building, Natural Place, Village, Animal, Plant, Album, Film, Written Work. As the input, 16-dimensional trainable embeddings are mapped from the dictionary of 69 raw characters (Conneau et al., 2016). The maximum length is set to 448 and longer inputs are truncated while the shorter inputs are padded. The input embeddings are first transformed with a 1-D convolutional block consisting 64 filters with kernel width of 3 and stride of 2. Then, 8 convolution blocks as in (Conneau et al., 2016) are applied, with 64, 64, 128, 128, 256, 256, 512 and 512 filters respectively. All use the kernel width of 3, and after each two layers, max pooling is applied with kernel width of 3 and a stride of 2. All convolutions are followed by batch normalization and the ReLU nonlinearity. Convolutional filters use weight normalization with parameter 0.00001. The last convolution block is followed by $\mathbf { k }$ -max pooling with $k { = } 8$ (Conneau et al., 2016). Finally, we apply two fully-connected layers with 1024 hidden units. In contrast to (Conneau et al., 2016), we also use layer normalization (Lei Ba et al., 2016) after fully-connected layers as we observe this leads to more stable training behavior. Keys and queries are mapped from the output using a fully-connected layer followed by the ReLU nonlinearity, where the attention size is $d _ { a t t } { = } 1 6$ . Values are mapped from the output using a fully-connected layer of $d _ { o u t } { = } 6 4$ units and the ReLU nonlinearity, followed by layer normalization. For the baseline encoder, initial learning rate is chosen as 0.0008 and exponential decay is applied with a rate of 0.9 applied every $^ { 8 \mathrm { k } }$ iterations. The model is trained for 212k iterations. For prototypical learning model with softmax attention, the initial learning rate is chosen as 0.0008 and exponential decay is applied with a rate of 0.9 applied every 8k iterations. The model is trained for 146k iterations. For prototypical learning model with sparsemax attention, the initial learning rate is chosen as 0.0005 and exponential decay is applied with a rate of 0.82 applied every $^ \mathrm { 8 k }$ iterations. The model is trained for $2 7 0 \mathrm { k }$ iterations. All models use a batch size of 128 and gradient clipping above 20. We do not apply any data augmentation.
+
+# C.3 TABULAR DATA
+
+# C.3.1 ADULT CENSUS INCOME
+
+There are two output classes: whether or not the annual income is above $\$ 50\mathrm { k }$ . Categorical categories such as the ‘marital-status’ are mapped to multi-hot representations. Continuous variables are used after a fixed normalization transformation. For ‘age’, the transformation first subtracts 50 and then divides by 30. For ‘fnlwgt’, the transformation first takes the log, and then subtracts 9, and then divides by 3. For ‘education-num’, the transformation first subtracts 6 and then divides by 6. For ‘hours-per-week’, the transformation first subtracts 50 and then divides by 50. For ‘capital-gain’ and ‘capital-loss’, the normalization takes the log, and then subtracts 5, and then divides by 5. The concatenated features are then mapped to a 64 dimensional vector using a fully-connected layer, followed by the ReLU nonlinearity. The base encoder uses an LSTM architecture, with 4 timesteps. At each timestep, 64-dimensional inputs are applied after a dropout with rate 0.5. The output of the last timestep is used after applying a dropout with rate 0.5. Keys and queries are mapped from this output using a fully-connected layer followed by the ReLU nonlinearity, where the attention size is $d _ { a t t } { = } 1 6$ . Values are mapped from the output using a fully-connected layer of $d _ { o u t } { = } 1 6$ units and the ReLU nonlinearity, followed by layer normalization. For the baseline encoder, the initial learning rate is chosen as 0.002 and exponential decay is applied with a rate of 0.9 applied every $2 \mathrm { k }$ iterations. The model is trained for $4 . 5 \mathrm { k }$ iterations. For the models with attention in prototypical learning framework, the initial learning rate is chosen as 0.0005 and exponential decay is applied with a rate of 0.92 applied every $2 \mathrm { k }$ iterations. The softmax attention model is trained for $1 3 . 5 \mathrm { k }$ iterations and the sparsemax attention model is trained for $1 1 . 5 \mathrm { k }$ iterations. For the model with sparsity regularization, the initial learning rate is 0.003 and exponential decay is applied with a rate of 0.7 applied every 2k iterations, and the model is trained for 7k iterations. All models use a batch size of 128 and gradient clipping above 20. We do not apply any data augmentation.
+
+# D ADDITIONAL PROTOTYPE EXAMPLES
+
+Fig. 10 exemplify prototypes for CIFAR-10. For most cases, we observe the similarity of discriminative features between inputs and prototypes. For example, the body figures of birds, the shape of tires, the face patterns of dogs, the body figures of frogs, the appearance of the background sky for planes, are among the features apparent in examples.
+
+Fig. 11 shows additional prototype examples for DBPedia dataset. Prototypes have very similar sentence structure, words and concepts, while categorizing the sentences into ontologies.
+
+Fig. 12 shows example prototypes for ISIC Melanoma. In some cases, we observe the commonalities between input and prototypes that distinguish melanoma cases such as the non-circular geometry or irregularly-notched borders (Jerant et al., 2000). Compared to other datasets, ISIC Melonama dataset yields lower interpretable prototype quality on average. We hypothesize this to be due to the perceptual difficulty of the problem as well as the insufficient encoder performance shown by the lower classification accuracy (despite the acceptable AUC).
+
+Fig. 13 shows more comparison examples for prototypical learning framework with sparsemax attention vs. representer point selection (Yeh et al., 2018) on Animals with Attributes dataset. For some cases, including chimpanzee, zebra, dalmatian and tiger, ProtoAttend yields perceptually very similar samples. The similarity of the chimpanzee body form and the background, zebra patterns, dalmatian pattern on the grass, and tiger pattern and head pose, are prominent. Representer point selection fails to capture such similarity features as effectively. On the other hand, for bat, otter and wolf, the results are somewhat less satisfying. The wing part of the bat, multiple count of the otters with the background, and the color and furry head of the wolf seem to be captured, but with less apparent similarity than some other possible samples from the dataset. Representer point selection method also cannot be claimed to be successful in these cases. Lastly, for leopard, ProtoAttend only yields one non-zero prototype (which is indeed statistically rare given the model and sparsity choices). The pattern of the leopard image seems relevant, but it is also not fully satisfying to observe a single prototype that is not perceptually more similar. All of the test examples in Fig. 13 are classified correctly with our framework and all of the shown prototypes are also from the correct classes.
+
+# E COMPARISON OF CONFIDENCE-CONTROLLED PREDICTION FOR SOFTMAX VS. SPARSEMAX
+
+Figs. 14 and 15 show the accuracy vs. ratio of samples for softmax vs. sparsemax attention without confidence regularization. The baseline accuracy (at $100 \%$ prediction ratio) is higher for softmax attention for some datasets, whereas higher for sparsemax for some others. On the other hand, higher number of prototypes yielded by softmax attention results in a wider range for confidence-controlled prediction trade-off.
+
+![](images/034410d775358765eda1d519c1daeb3e662c320a26531ac53b3f0968afc698cc.jpg)  
+Figure 10: Example inputs and corresponding prototypes for CIFAR-10.
+
+![](images/6346891a1d9baf573abbbbd4d63a7e341f1418424e30ef470b09fb170aba7b49.jpg)
+
+![](images/8405a0a63865bc2883fe05bdd2dc5a40b8924bc6967a30ca93582dba712b965f.jpg)  
+Figure 12: Example inputs and corresponding prototypes for ISIC Melanoma (with sparsemax attention).
+
+![](images/1e04bb9f13278fd7c431a54f690147d6dfac1fddfd666069c913eda4e5a2268f.jpg)  
+Figure 13: Relevant samples found by ProtoAttend with sparsemax attention vs. representer point selection (Yeh et al., 2018) for the examples from Supplementary Material of (Yeh et al., 2018).
+
+![](images/29459b15e38c3d80ce18d3c943ab234845c13b8b07ab710d6543051d3ce1bbce.jpg)  
+Figure 14: Accuracy vs. ratio of samples for (a) MNIST and (b) Fashion MNIST, for confidence levels between 0 and 0.999.
+
+![](images/fb2536ee6dff6378458471d6bd4cb7c0362afe86d2f456fe18025df8f28b7af6.jpg)  
+Figure 15: Accuracy vs. ratio of samples for (a) DBpedia and (b) Adult Census Income, for confidence levels between 0 and 0.999.
+
+As an impactful case study, we consider melanoma detection problem with ISIC dataset (ISIC, 2016) in Supplementary Material. In medical diagnosis, it is strongly desired to maintain a sufficiently-high prediction performance, potentially by verifying the decisions of an AI system by medical experts in the cases where the AI models are not confident. By refraining from some predictions, as shown in Fig. 16, we demonstrate unprecedentedly high AUC values without using transfer learning or highly-customized models (Haenssle et al., 2018).
+
+![](images/8dc8c38f0305b29206d04ceb30d3dccb5e0de500f65f9183364883d44ceea47f.jpg)  
+Figure 16: Area-under-curve (AUC) vs. ratio of samples for ISIC Melanoma with softmax attention, for confidence values ranging between 0 and 0.99.
+
+# F HUMAN USER STUDY ON THE USEFULNESS OF PROTOTYPES
+
+We perform a user study by asking humans how much an extra image helps in explaining the guessed class of the input, after showing what the trained network predicts for that input. We consider the Animals with Attributes dataset (exemplified in Fig. 13). We randomly pick test samples and assess how much showing the top prototype makes a difference. The results in Table 5 shows that ProtoAttend picks
+
+Table 5: Human ratings (mean score and $9 5 \%$ confidence interval) on how much an extra image helps guessing the class of the input.   
+
+<table><tr><td rowspan=1 colspan=1>Samplingmethod</td><td rowspan=1 colspan=1>Score (out of 5)</td></tr><tr><td rowspan=1 colspan=1>Top prototype by ProtoAttend</td><td rowspan=1 colspan=1>4.33± 0.09</td></tr><tr><td rowspan=1 colspan=1>Randomly sampled from the predicted class</td><td rowspan=1 colspan=1>3.97 ± 0.12</td></tr><tr><td rowspan=1 colspan=1>Randomlysampled fromanyclass</td><td rowspan=1 colspan=1>1.33± 0.09</td></tr></table>
+
+# G CONTROLLING SPARSITY VIA REGULARIZATION
+
+![](images/38c031db2ac9516eef1580214ba82ca130f21a0dfed84a3594d2181343386939.jpg)
+
+Figure 17: Number of training iterations vs. median number prototypes to explain $9 5 \%$ of the decision (in logarithmic scale), for Fashion-MNIST with softmax attention.
+
+Fig. 17 shows the impact of sparsity regularization coefficient on training. By varying the value of $\lambda _ { s p a r s e }$ , the number of prototypes can be efficiently controlled. For high values of sparsity regularization coefficient, the model gets stuck at a point where it is forced to make decision from a low number of prototypes before the encoder model is properly learned, hence typically yields considerably lower performance. We also observe sparsity mechanism via sparsemax attention to yield better performance than softmax attention with high sparsity regularization.
+
+# H PROTOTYPE QUALITY
+
+In general, the following scenarios may yield low prototype quality:
+
+1. Lack of related samples in the candidate database.   
+2. Perceptual difference between humans and encoders in determining discriminative features.   
+3. High intra-class variability that makes training difficult.   
+4. Imperfect encoder that cannot yield fully accurate representations of the input.   
+5. Insufficiency of relational attention to determine weights from queries and keys.   
+6. Inefficient decoupling between encoder & attention blocks and the final decision block.
+
+There can be problem-dependent fundamental limitations on (1)-(3), whereas (4)-(6) are raised by choices of models and losses and can be further improved. We leave the quantification of prototype quality using information-theoretic metrics or discriminative neural networks to future work.
+
+# I UNDERSTANDING MISCLASSIFICATION CASES
+
+One of the benefits of prototypical learning is insights into wrong decision cases. Fig. 18 exemplifies prototypes with wrong labels, that give insights about why the model is confused about a particular input (e.g. due to similarity of the visual patterns). Such insights can be actionable to improve the model performance, such as adding more training samples for the confusing classes or modifying the loss functions.
+
+![](images/5fd04f5881a72367af0b9121f5e1e20ac9c96b37d66e173012084d35766900b9.jpg)  
+Figure 18: Example prototypes with wrong labels for CIFAR-10.

parse/train/SJgdnAVKDH/SJgdnAVKDH.md

@@ -0,0 +1,328 @@
+# REVISITING SELF-TRAINING FOR NEURAL SEQUENCE GENERATION
+
+Junxian He∗ Carnegie Mellon University junxianh@cs.cmu.edu
+
+Jiatao $\mathbf { G u } ^ { * }$ , Jiajun Shen, Marc’Aurelio Ranzato Facebook AI Research, New York, NY {jgu,jiajunshen,ranzato}@fb.com
+
+# ABSTRACT
+
+Self-training is one of the earliest and simplest semi-supervised methods. The key idea is to augment the original labeled dataset with unlabeled data paired with the model’s prediction (i.e. the pseudo-parallel data). While self-training has been extensively studied on classification problems, in complex sequence generation tasks (e.g. machine translation) it is still unclear how self-training works due to the compositionality of the target space. In this work, we first empirically show that selftraining is able to decently improve the supervised baseline on neural sequence generation tasks. Through careful examination of the performance gains, we find that the perturbation on the hidden states (i.e. dropout) is critical for self-training to benefit from the pseudo-parallel data, which acts as a regularizer and forces the model to yield close predictions for similar unlabeled inputs. Such effect helps the model correct some incorrect predictions on unlabeled data. To further encourage this mechanism, we propose to inject noise to the input space, resulting in a “noisy” version of self-training. Empirical study on standard machine translation and text summarization benchmarks shows that noisy self-training is able to effectively utilize unlabeled data and improve the performance of the supervised baseline by a large margin.1
+
+# 1 INTRODUCTION
+
+Deep neural networks often require large amounts of labeled data to achieve good performance. However, acquiring labels is a costly process, which motivates research on methods that can effectively utilize unlabeled data to improve performance. Towards this goal, semi-supervised learning (Chapelle et al., 2009) methods that take advantage of both labeled and unlabeled data are a natural starting point. In the context of sequence generation problems, semi-supervised approaches have been shown to work well in some cases. For example, back-translation (Sennrich et al., 2015) makes use of the monolingual data on the target side to improve machine translation systems, latent variable models (Kingma et al., 2014) are employed to incorporate unlabeled source data to facilitate sentence compression (Miao & Blunsom, 2016) or code generation (Yin et al., 2018).
+
+In this work, we revisit a much older and simpler semi-supervised method, self-training (ST, Scudder (1965)), where a base model trained with labeled data acts as a “teacher” to label the unannotated data, which is then used to augment the original small training set. Then, a “student” model is trained with this new training set to yield the final model. Originally designed for classification problems, common wisdom suggests that this method may be effective only when a good fraction of the predictions on unlabeled samples are correct, otherwise mistakes are going to be reinforced (Zhu & Goldberg, 2009). In the field of natural language processing, some early work have successfully applied self-training to word sense disambiguation (Yarowsky, 1995) and parsing (McClosky et al., 2006; Reichart & Rappoport, 2007; Huang & Harper, 2009).
+
+However, self-training has not been studied extensively when the target output is natural language. This is partially because in language generation applications (e.g. machine translation) hypotheses are often very far from the ground-truth target, especially in low-resource settings. It is natural to
+
+# Algorithm 1 Classic Self-training
+
+1: Train a base model $f _ { \theta }$ on $L = \{ \pmb { x } _ { i } , \pmb { y } _ { i } \} _ { i = 1 } ^ { l }$   
+2: repeat   
+3: Apply $f _ { \theta }$ to the unlabeled instances $U$   
+4: Select a subset $S \subset \{ ( x , f _ { \pmb \theta } ( \pmb x ) ) | \pmb x \in U \}$   
+5: Train a new model $f _ { \theta }$ on $S \cup L$   
+6: until convergence or maximum iterations are reached
+
+ask whether self-training can be useful at all in this case. While Ueffing (2006) and Zhang & Zong (2016) explored self-training in statistical and neural machine translation, only relatively limited gains were reported and, to the best of our knowledge, it is still unclear what makes self-training work. Moreover, Zhang & Zong (2016) did not update the decoder parameters when using pseudo parallel data noting that “synthetic target parts may negatively influence the decoder model of NMT”.
+
+In this paper, we aim to answer two questions: (1) How does self-training perform in sequence generation tasks like machine translation and text summarization? Are “bad” pseudo targets indeed catastrophic for self-training? (2) If self-training helps improving the baseline, what contributes to its success? What are the important ingredients to make it work?
+
+Towards this end, we first evaluate self-training on a small-scale machine translation task and empirically observe significant performance gains over the supervised baseline (§3.2), then we perform a comprehensive ablation analysis to understand the key factors that contribute to its success (§3.3). We find that the decoding method to generate pseudo targets accounts for part of the improvement, but more importantly, the perturbation of hidden states – dropout (Hinton et al., 2012) – turns out to be a crucial ingredient to prevent self-training from falling into the same local optimum as the base model, and this is responsible for most of the gains. To understand the role of such noise in self-training, we use a toy experiment to analyze how noise effectively propagates labels to nearby inputs, sometimes helping correct incorrect predictions $( \ S 4 . 1 )$ . Motivated by this analysis, we propose to inject additional noise by perturbing also the input. Comprehensive experiments on machine translation and text summarization tasks demonstrate the effectiveness of noisy self-training.
+
+# 2 SELF-TRAINING
+
+Formally, in conditional sequence generation tasks like machine translation, we have a parallel dataset $L = \{ \pmb { x } _ { i } , \pmb { y } _ { i } \} _ { i = 1 } ^ { l }$ and a large unlabeled dataset $\boldsymbol { U } = \{ \mathbf { x } _ { j } \} _ { j = l + 1 } ^ { l + u }$ , where $| U | > | L |$ in most cases. As shown in Algorithm 1, classic self-training starts from a base model trained with parallel data $L$ , and iteratively applies the current model to obtain predictions on unlabeled instances $U$ , then it incorporates a subset of the pseudo parallel data $S$ to update the current model.
+
+There are two key factors: (1) Selection of the subset $S$ . $S$ is usually selected based on some confidence scores (e.g. log probability) (Yarowsky, 1995) but it is also possible for $S$ to be the whole pseudo parallel data (Zhu & Goldberg, 2009). (2) Combination of real and pseudo parallel data. A new model is often trained on the two datasets jointly as in back-translation, but this introduces an additional hyper-parameter to weigh the importance of the parallel data relative to the pseudo data (Edunov et al., 2018). Another way is to treat them separately – first we train the model on pseudo parallel data $S$ , and then fine-tune it on real data $L$ . In our preliminary experiments, we find that the separate training strategy with the whole pseudo parallel dataset (i.e. $\mathbf { \bar { \nabla } } S = \{ ( \mathbf { { \boldsymbol { x } } } , f _ { \theta } ( \mathbf { { \boldsymbol { x } } } ) ) | \mathbf { { \boldsymbol { x } } } \in U \} )$ produces better or equal performance for neural sequence generation while being simpler. Therefore, in the remainder of this paper we use this simpler setting. We include quantitative comparison regarding joint training, separate training, and pseudo-parallel data filtering in Appendix $\mathbf { B }$ , where separate training is able to match (or surpass) the performance of joint training.
+
+In self-training, the unsupervised loss $\mathcal { L } _ { U }$ from unlabeled instances is defined as:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { U } = - \mathbb { E } _ { { \pmb x } \sim p ( { \pmb x } ) } \mathbb { E } _ { { \pmb y } \sim p _ { \pmb \theta ^ { * } } ( { \pmb y } | { \pmb x } ) } \log p _ { \pmb \theta } ( { \pmb y } | { \pmb x } ) , } \end{array}
+$$
+
+where $p ( { \pmb x } )$ is the empirical data distribution approximated with samples from $S$ , $p _ { \pmb { \theta } } ( \pmb { y } | \pmb { x } )$ is the conditional distribution defined by the model. $\pmb { \theta } ^ { * }$ is the parameter from the last iteration (initially it
+
+<table><tr><td>Methods</td><td>PT</td><td>FT</td></tr><tr><td>baseline</td><td>1</td><td>15.6</td></tr><tr><td>ST (scratch)</td><td>16.8</td><td>17.9</td></tr><tr><td>ST (baseline)</td><td>16.5</td><td>17.5</td></tr></table>
+
+![](images/da636825754350288176394216a5f9a294d7de4d224c9b3ed76e0b57cc613735.jpg)  
+Figure 1: BLEU on WMT100K dataset from the supervised baseline and different self-training variants. We plot the results over 3 iterations. “ST” denotes self-training while “NST” denotes noisy self training.
+
+Table 1: Test tokenized BLEU on WMT100K. Self-training results are from the first iteration. “Scratch” denotes that the system is initialized randomly and trained from scratch, while “baseline” means it is initialized with the baseline model.
+
+is set as the parameter of the supervised baseline), and fixed within the current iteration. Eq. 1 reveals the connection between self-training and entropy regularization (Grandvalet & Bengio, 2005). In the context of classification, self-training can be understood from the view of entropy regularization (Lee, 2013), which favors a low-density separation between classes, a commonly assumed prior for semi-supervised learning (Chapelle & Zien, 2005).
+
+# 3 A CASE STUDY ON MACHINE TRANSLATION
+
+To examine the effectiveness of self-training on neural sequence generation, we start by analyzing a machine translation task. We then perform ablation analysis to understand the contributing factors of the performance gains.
+
+# 3.1 SETUP
+
+We work with the standard WMT 2014 English-German dataset consisting of about 3.9 million training sentence pairs after filtering long and imbalanced pairs. Sentences are encoded using 40K byte-pair codes (Sennrich et al., 2016). As a preliminary experiment, we randomly sample 100K sentences from the training set to train the model and use the remaining English sentences as the unlabeled monolingual data. For convenience, we refer to this dataset as WMT100K. Such synthetic setting allows us to have high-quality unlabeled data to verify the performance of self-training. We train with the Base Transformer architecture (Vaswani et al., 2017) and dropout rate at 0.3. Full training and optimization parameters can be found in Appendix A.1. All experiments throughout this paper including the transformer implementation are based on the fairseq toolkit (Ott et al., 2019), and all results are in terms of case-sensitive tokenized BLEU (Papineni et al., 2002). We use beam search decoding (beam size 5) to create the pseudo targets and to report BLEU on test set.
+
+# 3.2 OBSERVATIONS
+
+In Figure 1, we use green bars to show the result of applying self-training for three iterations. We include both (1) pseudo-training $( P T )$ : the first step of self-training where we train a new model (from scratch) using only the pseudo parallel data generated by the current model, and (2) finetuning $( F T )$ : the fine-tuned system using real parallel data based on the pretrained model from the PT step. Note that in the fine-tuning step the system is re-initialized from scratch. Surprisingly, we find that the pseudo-training step at the first iteration is able to improve BLEU even if the model is only trained on its own predictions, and fine-tuning further boosts the performance. The test BLEU keeps improving over the first three iterations, until convergence to outperform the initial baseline by 3 BLEU points.
+
+<table><tr><td>Methods</td><td>PT</td><td>FT</td></tr><tr><td>baseline baseline (w/o dropout)</td><td>1</td><td>15.6 5.2</td></tr><tr><td>ST (beam search,w/ dropout)</td><td>1 16.5</td><td>17.5</td></tr><tr><td>ST (sampling, w/ dropout)</td><td>16.1</td><td>17.0</td></tr><tr><td>ST (beam search,w/o dropout)</td><td>15.8</td><td>16.3</td></tr><tr><td>ST (sampling, w/o dropout)</td><td>15.5</td><td>16.0</td></tr><tr><td>Noisy ST (beam search, w/o dropout)</td><td></td><td>17.9</td></tr><tr><td>Noisy ST (beam search, w/ dropout)</td><td>15.8 16.6</td><td>19.3</td></tr></table>
+
+Table 2: Ablation study on WMT100K data. For ST and noisy ST, we initialize the model with the baseline and results are from one single iteration. Dropout is varied only in the PT step, while dropout is always applied in FT step. Different decoding methods refer to the strategy used to create the pseudo target. At test time we use beam search decoding for all models.
+
+This behaviour is unexpected because no new information seems to be injected during this iterative process – target sentences of the monolingual data are from the base model’s predictions, thus translation errors are likely to remain, if not magnified. This is different from back-translation where new knowledge may originate from an additional backward translation model and real monolingual targets may help the decoder generate more fluent sentences.
+
+One straightforward hypothesis is that the added pseudo-parallel data might implicitly change the training trajectory towards a (somehow) better local optimum, given that we train a new model from scratch at each iteration. To rule out this hypothesis, we perform an ablation experiment and initialize $\pmb { \theta }$ from the last iteration (i.e. $\pmb { \theta } ^ { * }$ ). Formally, based on Eq. 1 we have:
+
+$$
+\nabla _ { \pmb { \theta } } \mathcal { L } _ { U } | _ { \pmb { \theta = \theta } ^ { \ast } } = - \mathbb { E } _ { \pmb { x } \sim p ( \pmb { x } ) } \left[ \nabla _ { \pmb { \theta } } \mathbb { E } _ { \pmb { y } \sim p _ { \theta ^ { \ast } } ( \pmb { y } | \pmb { x } ) } \log p _ { \theta } ( \pmb { y } | \pmb { x } ) | _ { \pmb { \theta = \theta } ^ { \ast } } \right] = 0 ,
+$$
+
+because the conditional log likelihood is maximized when $p _ { \pmb { \theta } } ( \pmb { y } | \pmb { x } )$ matches the underlying data distribution $p _ { \pmb { \theta } ^ { \ast } } ( \pmb { y } | \pmb { x } )$ . Therefore, the parameter $\pmb \theta$ should not (at least not significantly) change if we initialize it with $\pmb { \theta } ^ { * }$ from the last iteration.
+
+Table 1 shows the comparison results of these two initialization schemes at the first iteration. Surprisingly, continuing training from the baseline model also yields an improvement of 1.9 BLEU points, comparable to initializing from random. While stochastic optimization introduces randomness in the training process, it is startling that continuing training gives such a non-trivial improvement. Next, we investigate the underlying reasons for this.
+
+# 3.3 THE SECRET BEHIND SELF-TRAINING
+
+To understand why continuing training contradicts Eq. 2 and improves translation performance, we examine possible discrepancies between our assumptions and the actual implementation, and formulate two new hypotheses:
+
+H1. Decoding Strategy. According to this hypothesis, the gains come from the use of beam search for decoding unlabeled data. Since our focus is a sequence generation task, we decode $\textbf {  { y } }$ with beam search to approximate the expectation in $\mathbb { E } _ { { \pmb { y } } \sim p _ { \pmb { \theta } ^ { * } } ( { \pmb { y } } | { \pmb x } ) } \log { \bar { p } _ { \pmb { \theta } } ( { \pmb { y } } | { \pmb x } ) }$ , yielding a biased estimate, while sampling decoding would result in an unbiased Monte Carlo estimator. The results in Table 2 demonstrate that the performance drops by 0.5 BLEU when we change the decoding strategy to sampling, which implies that beam search does contribute a bit to the performance gains. This phenomenon makes sense intuitively since beam search tends to generate higher-quality pseudo targets than sampling, and the subsequent cross-entropy training might benefit from implicitly learning the decoding process. However, the decoding strategy hypothesis does not fully explain it, as we still observe a gain of 1.4 BLEU points over the baseline from sampling decoding with dropout.
+
+H2. Dropout (Hinton et al., 2012). Eq. 1 and Eq. 2 implicitly ignore a (seemingly) small difference between the model used to produce the pseudo targets and the model used for training: at test/decoding time the model does not use dropout while at training time dropout noise is injected in the model hidden states. At training time, the model is forced to produce the same (pseudo) targets given the same set of inputs and the same parameter set but various noisy versions of the hidden states. The conjecture is that the additional expectation over dropout noise renders Eq. 2 false. To verify this, we remove dropout in the pseudo training step2. The results in Table 2 indicate that without dropout the performance of beam search decoding drops by 1.2 BLEU, just 0.7 BLEU higher than the baseline. Moreover, the pseudo-training performance of sampling without dropout is almost the same as the baseline, which finally agrees with our intuitions from Eq. 2.
+
+In summary, Table 2 suggests that beam-search decoding contributes only partially to the performance gains, while the implicit perturbation – dropout – accounts for most of it. However, it is still mysterious why such perturbation results in such large performance gains. If dropout is meant to avoid overfitting and fit the target distribution better in the pseudo-training step, why does it bring advantages over the baseline given that the target distribution is from the baseline model itself ? This is the subject of the investigation in the next section.
+
+# 4 NOISE IN SELF-TRAINING
+
+# 4.1 THE ROLE OF NOISE
+
+One hypothesis as to why noise (perturbation) is beneficial for self-training, is that it enforces local smoothness for this task, that is, semantically similar inputs are mapped to the same or similar targets. Since the assumption that similar input should ideally produce similar target largely holds for most tasks in practice, this smoothing effect of pseudo-training step may provide a favorable regularization for the subsequent finetuning step. Unlike standard regularization in supervised training which is local to the real parallel data, self-training smooths the data space covered by the additional and much larger monolingual data.
+
+Table 3: Results on the toy sum dataset. For ST and noisy ST, smoothness (↓) and symmetric (↓) results are from the pseudo-training step, while test errors (↓) are from fine-tuning, all at the first iteration.   
+
+<table><tr><td>Methods</td><td>smoothness</td><td>symmetric</td><td>error</td></tr><tr><td>baseline</td><td>9.1</td><td>9.8</td><td>7.6</td></tr><tr><td>ST</td><td>8.2</td><td>9.0</td><td>6.2</td></tr><tr><td>noisy ST</td><td>7.3</td><td>8.2</td><td>4.5</td></tr></table>
+
+To verify this hypothesis more easily, we work with the toy task of summing two integers in the range 0 to 99. We concatenate the two integers and view them as a sequence of digits, the sum is also predicted at the digit level, thus this is still a sequence to sequence task. There are 10000 possible data points in the entire space, and we randomly sample 250 instances for training,3 100 for validation, 5000 for test, and 4000 as the unlabeled data. Test errors are computed as the absolute difference between the predicted integer and the ground-truth integer. We use an LSTM model to tackle this task. We perform self-training for one iteration on this toy sum dataset and initialize the model with the base model to rule out differences due to the initialization. Setup details are in Appendix A.1.
+
+For any integer pair $( x _ { 1 } , x _ { 2 } )$ , we measure local smoothness as the standard deviation of the predictions in a $3 \times 3$ neighborhood of $( x _ { 1 } , x _ { 2 } )$ . These values are averaged over all the 10000 points to obtain the overall smoothness. We compare smoothness between baseline and ST pseudo-training in Table 3. To demonstrate the effect of smoothing on the fine-tuning step, we also report test errors after fine-tuning. We observe that ST pseudo-training attains better smoothness, which helps reducing test errors in the subsequent fine-tuning step.
+
+One natural question is whether we could further improve performance by encouraging even lower smoothness value, although there is a clear trade-off, as a totally smooth model that outputs a constant value is also a bad predictor. One way to decrease smoothness is by increasing the dropout probability in the pseudo-training step, but a large dropout (like 0.5) makes the model too unstable and slow at converging. Therefore, we consider a simple model-agnostic perturbation process – perturbing the input, which we refer to as noisy self-training (noisy ST).
+
+![](images/5d0d65d405c12883f059b336b7d310b3cef8725e1bfc459455866d0394e0bdcb.jpg)  
+Figure 2: Two examples of error heat map on the toy sum dataset that shows the effect of smoothness. The left panel of each composition is from the baseline, and the right one is from the pseudo-training step at the first iteration. $x$ and $y$ axes represent the two input integers. Deeper color represent larger errors.
+
+# 4.2 NOISY SELF-TRAINING
+
+If we perturb the input during the pseudo-training step, then Eq. 1 would be modified to:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { U } = - \mathbb { E } _ { { \pmb x } ^ { \prime } \sim g ( { \pmb x } ) , { \pmb x } \sim p ( { \pmb x } ) } \mathbb { E } _ { { \pmb y } \sim p _ { \theta ^ { * } } ( { \pmb y } | { \pmb x } ) } \log p _ { \theta } ( { \pmb y } | { \pmb x } ^ { \prime } ) , } \end{array}
+$$
+
+where $g ( { \pmb x } )$ is a perturbation function. Note that we apply both input perturbation and dropout in the pseudo-training step for noisy ST throughout the paper, but include ablation analysis in $\ S 4 . 3$ . We first validate noisy ST in the toy sum task. We shuffle the two integers in the input as the perturbation function. Such perturbation is suitable for this task since it would help the model learn the commutative law as well. To check that, we also measure the symmetry of the output space. Specifically, for any point $( x _ { 1 } , x _ { 2 } )$ , we compute $| f ( x _ { 1 } , x _ { 2 } ) - f ( x _ { 2 } , x _ { 1 } ) |$ and average it over all the points. Both smoothness and symmetry values are reported in Table 3. While we do not explicitly perturb the input at nearby integers, the shuffling perturbation greatly improves the smoothness metric as well. Furthermore, predictions are more symmetric and test errors are reduced.
+
+In order to illustrate the effect of smoothness, in Figure 2 we show two examples of error heat map.4 When a point with large error is surrounded by points with small errors, the labels might propagate due to smoothing and its error is likely to become smaller, resulting in a “self-correcting” behaviour, as demonstrated in the left example of Figure 2. However, the prediction of some points might become worse due to the opposite phenomenon too, as shown in the right example of Figure 2. Therefore, the smoothing effect by itself does not guarantee a performance gain in the pseudotraining step, but fine-tuning benefits from it and seems to consistently improve the baseline in all datasets we experiment with.
+
+# 4.3 OBSERVATIONS ON MACHINE TRANSLATION
+
+Next, we apply noisy self-training to the more realistic WMT100 translation task. We try two different perturbation functions: (1) Synthetic noise as used in unsupervised MT (Lample et al., 2018), where the input tokens are randomly dropped, masked, and shuffled. We use the default noising parameters as in unsupervised MT but study the influence of noise level in $\ S 5 . 4$ . (2) Paraphrase. We translate the source English sentences to German and translate it back to obtain a paraphrase as the perturbation. Figure 1 shows the results over three iterations. Noisy ST (NST) greatly outperforms the supervised baseline by over 6 BLEU points and normal ST by 3 BLEU points, while synthetic noise does not exhibit much difference from paraphrasing. Since synthetic noise is much simpler and more general, in the remaining experiments we use synthetic noise unless otherwise specified.
+
+Next, we report an ablation analysis of noisy ST when removing dropout at the pseudo-training step in Table 2. Noisy ST without dropout improves the baseline by 2.3 BLEU points and is comparable to normal ST with dropout. When combined together, noisy ST with dropout produces another 1.4 BLEU improvement, indicating that the two perturbations are complementary.
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">WMT English-German</td><td colspan="3">FloRes English-Nepali</td></tr><tr><td>100K (+3.8M mono)</td><td>3.9M (+20M mono)</td><td>En-Origin</td><td>Ne-Origin</td><td>Overall</td></tr><tr><td>baseline</td><td>15.6</td><td>28.3</td><td>6.7</td><td>2.3</td><td>4.8</td></tr><tr><td>BT</td><td>20.5</td><td>1</td><td>8.2</td><td>4.5</td><td>6.5</td></tr><tr><td>noisy ST</td><td>21.4</td><td>29.3</td><td>8.9</td><td>3.5</td><td>6.5</td></tr></table>
+
+Table 4: Results on two machine translation datasets. For WMT100K, we use the remaining 3.8M English and German sentences from training data as unlabeled data for noisy ST and BT, respectively.
+
+# 5 EXPERIMENTS
+
+Our experiments below are designed to examine whether the noisy self-training is generally useful across different sequence generation tasks and resource settings. To this end, we conduct experiments on two machine translation datasets and one text summarization dataset to test the effectiveness under both high-resource and low-resource settings.
+
+# 5.1 GENERAL SETUP
+
+We run noisy self-training for three iterations or until performance converges. The model is trained from scratch in the pseudo-training step at each iteration since we found this strategy to work slightly better empirically. Full model and training details for all the experiments can be found in Appendix A.1. In some settings, we also include back-translation (BT, Sennrich et al., 2015) as a reference point, since this is probably the most successful semi-supervised learning method for machine translation. However, we want to emphasize that BT is not directly comparable to ST since they use different resources (ST utilizes the unlabeled data on the source side while BT leverages target monolingual data) and use cases. For example, BT is not very effective when we translate English to extremely low-resource languages where there is almost no in-domain target monolingual data available. We follow the practice in (Edunov et al., 2018) to implement BT where we use unrestricted sampling to translate the target data back to the source. Then, we train the real and pseudo parallel data jointly and tune the upsampling ratio of real parallel data.
+
+# 5.2 MACHINE TRANSLATION
+
+We test the proposed noisy self-training on a high-resource translation benchmark: WMT14 EnglishGerman and a low-resource translation benchmark: FloRes English-Nepali.
+
+• WMT14 English-German: In addition to WMT100K, we also report results with all 3.9M   
+training examples. For WMT100K we use the Base Transformer architecture, and the remaining parallel data as the monolingual data. For the full setting, we use the Big Transformer architecture (Vaswani et al., 2017) and randomly sample 20M English sentences from the News Crawl corpus for noisy ST.   
+FloRes English-Nepali: We evaluate noisy self-training on a low-resource machine translation dataset FloRes (Guzman et al., 2019) from English (en) to Nepali (ne), where we have 560K train- ´ ing pairs and a very weak supervised system that attains BLEU smaller than 5 points. For this dataset we have 3.6M Nepali monolingual instances in total (for BT) but 68M English Wikipedia   
+sentences.5 We randomly sample 5M English sentences for noisy ST. We use the same transformer architecture as in (Guzman et al., 2019). ´
+
+The overall results are shown in Table 4. For almost all cases in both datasets, the noisy ST outperforms the baselines by a large margin $\mathrm { . 1 \sim 5 }$ BLEU scores), and we see that noisy ST still improves the baseline even when this is very weak.
+
+Effect of Domain Mismatch. Test sets of the FloRes benchmark were built with mixed originaltranslationese – some sentences are from English sources and some are from Nepali sources. Intuitively, English monolingual data should be more in-domain with English-origin sentences and
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">100K (+3.7M mono)</td><td colspan="3">640K (+3.2M mono)</td><td colspan="3">3.8M(+4M mono)</td></tr><tr><td>R1</td><td>R2</td><td>RL</td><td>R1</td><td>R2</td><td>RL</td><td>R1</td><td>R2</td><td>RL</td></tr><tr><td>MASS (Song et al., 2019)*</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>38.7</td><td>19.7</td><td>36.0</td></tr><tr><td>baseline</td><td>30.4</td><td>12.4</td><td>27.8</td><td>35.8</td><td>17.0</td><td>33.2</td><td>37.9</td><td>19.0</td><td>35.2</td></tr><tr><td>BT</td><td>32.2</td><td>13.8</td><td>29.6</td><td>37.3</td><td>18.4</td><td>34.6</td><td>1</td><td>一</td><td>一</td></tr><tr><td>noisy ST</td><td>34.1</td><td>15.6</td><td>31.4</td><td>36.6</td><td>18.2</td><td>33.9</td><td>38.6</td><td>19.5</td><td>35.9</td></tr></table>
+
+Table 5: Rouge scores on Gigaword datasets. For the 100K setting we use the remaining 3.7M training data as unlabeled instances for noisy ST and BT. In the $3 . 8 \mathbf { M }$ setting we use 4M unlabeled data for noisy ST. Stared entry $( * )$ denotes that the system uses a much larger dataset for pretraining.
+
+![](images/b08dfd4e8a0d8b7b4ecf148a482647b43eb4bd984377e7fded1224f6a549cbc2.jpg)  
+Figure 3: Analysis of noisy self-training on WMT English-German dataset, demonstrating the effect of parallel data size, monolingual data size, and noise level.
+
+Nepali monolingual data should help more for Nepali-origin sentences. To demonstrate this possible domain-mismatch effect, in Table 4 we report BLEU on the two different test sets separately.6 As expected, ST is very effective when the source sentences originate from English.
+
+Comparison to Back-Translation. Table 4 shows that noisy ST is able to beat BT on WMT100K and on the en-origin test set of FloRes. In contrast, BT is more effective on the ne-origin test set according to BLEU, which is not surprising as the ne-origin test is likely to benefit more from Nepali than English monolingual data.
+
+# 5.3 TEXT SUMMARIZATION
+
+We further evaluate noisy self-training on the Gigaword summarization dataset (Rush et al., 2015) that has $3 . 8 \mathbf { M }$ training sentences. We encode the data with 30K byte-pair codes and use the Base Transformer architecture. Similar to the setting of WMT100K, for Gigaword we create two settings where we sample 100K or 640K training examples and use the remaining as unlabeled data to compare with BT. We also consider the setting where all the 3.8M parallel samples are used and we mine in-domain monolingual data by revisiting the original preprocessing procedure7 and using the ${ \sim } 4 \mathbf { M }$ samples that Rush et al. (2015) disregarded because they had low-quality targets. We report ROUGE scores (Lin, 2004) in Table 5. Noisy ST consistently outperforms the baseline in all settings, sometimes by a large margin (100K and 640K). It outperforms BT with 100K parallel data but underperforms with 640K parallel data. We conjecture that BT is still effective in this case because the task is still somewhat symmetric as Gigaword mostly contains short sentences and their compressed summaries. Notably, noisy ST in the full setting approaches the performance of state-of-the-art systems which use much larger datasets for pretraining (Song et al., 2019).
+
+# 5.4 ANALYSIS
+
+In this section, we focus on the WMT English-German dataset to examine the effect of three factors on noisy self-training: the size of the parallel dataset, the size of the monolingual dataset, and the noise level. All the noisy ST results are after the fine-tuning step.
+
+Parallel data size. We fix the monolingual data size as 20M from News Crawl dataset, and vary the parallel data size as shown in Figure 3(a). We use a small LSTM model for 10K, Base Transformer for 100K/640K, and Big Transformer for 3.9M.8 Noisy ST is repeated for three iterations. We see that in all cases noisy ST is able to improve upon the baseline, while the performance gain is larger for intermediate value of the size of the parallel dataset, as expected.
+
+Monolingual data size. We fix the parallel data size to 100K samples, and use the rest 3.8M English sentences from the parallel data as monolingual data. We sample from this set 100K, 500K, 1.5M, and $3 . 8 \mathbf { M }$ sentences. We also include another point that uses 20M monolingual sentences from a subset of News Crawl dataset. We report performance at the first iteration of noisy ST. Figure 3(b) illustrates that the performance keeps improving as the monolingual data size increases, albeit with diminishing returns.
+
+Noise level. We have shown that noisy ST outperforms ST, but intuitively larger noise must not always be better since at some point it may destroy all the information present in the input. We adopt the WMT100K setting with 100K parallel data and 3.8M monolingual data, and set the word blanking probability in the synthetic noise (Lample et al., 2018) to 0.2 (default number), 0.4, 0.6, and 0.8. We also include the baseline ST without any synthetic noise. Figure 3(c) demonstrates that performance is quite sensitive to noise level, and that intermediate values work best. It is still unclear how to select the noise level a priori, besides the usual hyper-parameter search to maximize BLEU on the validation set.
+
+# 5.5 NOISE PROCESS ON PARALLEL DATA ONLY
+
+In this section, we justify whether the proposed noisy self-training process would help the supervised baseline alone without the help of any monolingual data. Similar to the training process on the monolingual data, we first train the model on the noisy source data (pseudo-training), and then finetune it on clean parallel data. Different from using monolingual data, there are two variations here in the “pseudo-training” step: we can either train with the fake target predicted by the model as on monolingual data, or train with the real target paired with noisy source. We denote them as “parallel $^ +$ fake target” and “parallel $^ +$ real target” respectively, and report the performance on WMT100K in Table 6. We use the same synthetic noise as used in previous experiments.
+
+When applying the same noise process to parallel data using fake target, the smoothing effect is not significant since it is restricted into the limited parallel data space, producing marginal improvement over the baseline $( + 0 . 4$ BLEU). As a comparison, 100K monolingual data produces $+ 1 . 0$ BLEU and the effect is enhanced when we increase the monolingual data to $3 . 8 \mathbf { M }$ , which leads to $+ 3 . 7$ BLEU. Interestingly, pairing the noisy source with real target results in much worse performance than the baseline (-4.3 BLEU), which implies that the use of fake target predicted by the model (i.e. distillation) instead of real target is important for the success of noisy self-training, at least in the case where parallel data size is small. Intuitively, the distilled fake target is simpler and relatively easy for the model to fit, but the real target paired with noisy source makes learning even harder than training with real target and real source, which might lead to a bad starting point for fine-tuning. This issue would be particularly severe when the parallel data size is small, in that case the model would have difficulties to fit real target even with clean source.
+
+# 6 RELATED WORK
+
+Self-training belongs to a broader class of “pseudo-label” semi-supervised learning approaches. These approaches all learn from pseudo labels assigned to unlabelled data, with different methods on how to assign such labels. For instance, co-training (Blum & Mitchell, 1998) learns models on two independent feature sets of the same data, and assigns confident labels to unlabeled data from one of the models. Co-training reduces modeling bias by taking into account confidence scores from two models. In the same spirit, democratic co-training (Zhou & Goldman, 2004) or tri-training (Zhou & Li, 2005) trains multiple models with different configurations on the same data feature set, and a subset of the models act as teachers for others.
+
+<table><tr><td>Methods</td><td>PT</td><td>FT</td></tr><tr><td> parallel baseline</td><td>1</td><td>15.6</td></tr><tr><td>noisy ST,100K mono + fake target</td><td>10.2</td><td>16.6</td></tr><tr><td> noisy ST, 3.8M mono + fake target</td><td>16.6</td><td>19.3</td></tr><tr><td>noisy ST,100K parallel +real target noisy ST,10OK parallel + fake target</td><td>6.7 10.4</td><td>11.3 16.0</td></tr></table>
+
+Table 6: Results on WMT100K data. All results are from one single iteration. “Parallel $^ +$ real/fake target” denotes the noise process applied on parallel data but using real/fake target in the “pseudotraining” step. “Mono $^ +$ fake target” is the normal noisy self-training process described in previous sections.
+
+Another line of more recent work perturb the input or feature space of the student’s inputs as data augmentation techniques. Self-training with dropout or noisy self-training can be viewed as an instantiation of this. These approaches have been very successful on classification tasks (Rasmus et al., 2015; Miyato et al., 2017; Laine & Aila, 2017; Miyato et al., 2018; Xie et al., 2019) given that a reasonable amount of predictions of unlabeled data (at least the ones with high confidence) are correct, but their effect on language generation tasks is largely unknown and poorly understood because the pseudo language targets are often very different from the ground-truth labels. Recent work on sequence generation employs auxiliary decoders (Clark et al., 2018) when processing unlabeled data, overall showing rather limited gains.
+
+# 7 CONCLUSION
+
+In this paper we revisit self-training for neural sequence generation, and show that it can be an effective method to improve generalization, particularly when labeled data is scarce. Through a comprehensive ablation analysis and synthetic experiments, we identify that noise injected during self-training plays a critical role for its success due to its smoothing effect. To encourage this behaviour, we explicitly perturb the input to obtain a new variant of self-training, dubbed noisy selftraining. Experiments on machine translation and text summarization demonstrate the effectiveness of this approach in both low and high resource settings.
+
+# ACKNOWLEDGEMENTS
+
+We want to thank Peng-Jen Chen for helping set up the FloRes experiments, and Michael Auli, Kyunghyun Cho, and Graham Neubig for insightful discussion about this project.
+
+# REFERENCES
+
+Avrim Blum and Tom Mitchell. Combining labeled and unlabeled data with co-training. In Proceedings of the eleventh annual conference on Computational learning theory, pp. 92–100. Citeseer, 1998.
+
+Olivier Chapelle and Alexander Zien. Semi-supervised classification by low density separation. In Proceedings of AISTATS, 2005.
+
+Olivier Chapelle, Bernhard Scholkopf, and Alexander Zien. Semi-supervised learning (chapelle, o. et al., eds.; 2006)[book reviews]. IEEE Transactions on Neural Networks, 20(3):542–542, 2009.
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+Kevin Clark, Minh-Thang Luong, Christopher D Manning, and Quoc V Le. Semi-supervised sequence modeling with cross-view training. In Proceedings of EMNLP, 2018.
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+Sergey Edunov, Myle Ott, Michael Auli, and David Grangier. Understanding back-translation at scale. In Proceedings of EMNLP, 2018.
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+Yves Grandvalet and Yoshua Bengio. Semi-supervised learning by entropy minimization. In Proceedings of NeurIPS, 2005.
+
+Francisco Guzman, Peng-Jen Chen, Myle Ott, Juan Pino, Guillaume Lample, Philipp Koehn, ´ Vishrav Chaudhary, and Marc’Aurelio Ranzato. The FLoRes evaluation datasets for low-resource machine translation: Nepali-english and sinhala-english. In Proceedings of EMNLP, 2019.
+
+Geoffrey E Hinton, Nitish Srivastava, Alex Krizhevsky, Ilya Sutskever, and Ruslan R Salakhutdinov. Improving neural networks by preventing co-adaptation of feature detectors. arXiv preprint arXiv:1207.0580, 2012.
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+Zhongqiang Huang and Mary Harper. Self-training pcfg grammars with latent annotations across languages. In Proceedings of EMNLP, 2009.
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+Dong-Hyun Lee. Pseudo-label: The simple and efficient semi-supervised learning method for deep neural networks. In Workshop on Challenges in Representation Learning, ICML, 2013.
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+Chin-Yew Lin. Rouge: A package for automatic evaluation of summaries. In Text summarization branches out, pp. 74–81, 2004.
+
+David McClosky, Eugene Charniak, and Mark Johnson. Effective self-training for parsing. In Proceedings of NAACL, 2006.
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+Yishu Miao and Phil Blunsom. Language as a latent variable: Discrete generative models for sentence compression. In Proceedings of EMNLP, 2016.
+
+Takeru Miyato, Andrew M Dai, and Ian Goodfellow. Adversarial training methods for semisupervised text classification. In Proceedings of ICLR, 2017.
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+Takeru Miyato, Shin-ichi Maeda, Masanori Koyama, and Shin Ishii. Virtual adversarial training: a regularization method for supervised and semi-supervised learning. IEEE transactions on pattern analysis and machine intelligence, 41(8):1979–1993, 2018.
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+Myle Ott, Sergey Edunov, Alexei Baevski, Angela Fan, Sam Gross, Nathan Ng, David Grangier, and Michael Auli. fairseq: A fast, extensible toolkit for sequence modeling. In Proceedings of NAACL (Demo Track), 2019.
+
+Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. BLEU: a method for automatic evaluation of machine translation. In Proceedings of ACL, 2002.
+
+Antti Rasmus, Mathias Berglund, Mikko Honkala, Harri Valpola, and Tapani Raiko. Semisupervised learning with ladder networks. In Proceedings of NeurIPS, 2015.
+
+Roi Reichart and Ari Rappoport. Self-training for enhancement and domain adaptation of statistical parsers trained on small datasets. In Proceedings of ACL, 2007.
+
+Alexander M Rush, Sumit Chopra, and Jason Weston. A neural attention model for abstractive sentence summarization. In Proceedings of EMNLP, 2015.
+
+H Scudder. Probability of error of some adaptive pattern-recognition machines. IEEE Transactions on Information Theory, 11(3):363–371, 1965.
+
+Rico Sennrich, Barry Haddow, and Alexandra Birch. Improving neural machine translation models with monolingual data. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics, pp. 86–96, 2015.
+
+Rico Sennrich, Barry Haddow, and Alexandra Birch. Neural machine translation of rare words with subword units. In Proceedings of ACL, 2016.
+
+Kaitao Song, Xu Tan, Tao Qin, Jianfeng Lu, and Tie-Yan Liu. MASS: Masked sequence to sequence pre-training for language generation. In Proceedings of ICML, 2019.
+
+Nicola Ueffing. Using monolingual source-language data to improve mt performance. In IWSLT, 2006.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Proceedings of NeurIPS, 2017.
+
+Qizhe Xie, Zihang Dai, Eduard Hovy, Minh-Thang Luong, and Quoc V Le. Unsupervised data augmentation. arXiv preprint arXiv:1904.12848, 2019.
+
+David Yarowsky. Unsupervised word sense disambiguation rivaling supervised methods. In Proceedings of ACL, 1995.
+
+Pengcheng Yin, Chunting Zhou, Junxian He, and Graham Neubig. StructVAE: Tree-structured latent variable models for semi-supervised semantic parsing. In Proceedings of EMNLP, 2018.
+
+Jiajun Zhang and Chengqing Zong. Exploiting source-side monolingual data in neural machine translation. In Proceedings of EMNLP, 2016.
+
+Yan Zhou and Sally Goldman. Democratic co-learning. In 16th IEEE International Conference on Tools with Artificial Intelligence, pp. 594–602. IEEE, 2004.
+
+Zhi-Hua Zhou and Ming Li. Tri-training: Exploiting unlabeled data using three classifiers. IEEE Transactions on Knowledge & Data Engineering, (11):1529–1541, 2005.
+
+Xiaojin Zhu and Andrew B Goldberg. Introduction to semi-supervised learning. Synthesis lectures on artificial intelligence and machine learning, 3(1):1–130, 2009.
+
+# A EXPERIMENTS DETAILS
+
+# A.1 SETUP DETAILS
+
+For all experiments, we optimize with Adam (Kingma & Ba, 2014) using $\beta _ { 1 } = 0 . 9 , \beta _ { 2 } = 0 . 9 8 , \epsilon =$ $1 e - 8$ . All implementations are based on fairseq (Ott et al., 2019), and we basically use the same learning rate schedule and label smoothing as in fairseq examples to train the transformers.9 Except for the toy sum dataset which we runs on a single GPU and each batch contains 32 examples, all other experiments are run on 8 GPUs with an effective batch size of 33K tokens. All experiments are validated with loss on the validation set. For self-training or noisy self-training, the pseudo-training takes 300K synchronous updates while the fine-tuning step takes 100K steps.
+
+We use the downloading and preprocessing scripts in fairseq to obtain the WMT 2014 EnglishGerman dataset,10 which hold out a small fraction of the original training data as the validation set.
+
+The model architecture for the toy sum dataset is a single-layer LSTM with word embedding size 32, hidden state size 32, and dropout rate 0.3. The model architecture of WMT10K baseline in Figure 3(a) is a single layer LSTM with word embeddings size 256, hidden state size 256, and dropout rate 0.3.
+
+# A.2 JUSTIFICATION OF THE WMT100K BASELINE
+
+We provide more details and evidence to show that our baseline model on WMT100K dataset is trained properly. In all the experiments on WMT100K dataset including baseline and self-training ones, we use Adam optimizer with learning rate 0.0005, which is defaulted in fairseq. We do not use early stop during training but select the best model in terms of the validation loss. We train with 30K update steps for the baseline model and (300K pseudo-training $+ \ 1 0 0 \mathrm { K }$ fine-tuning) update steps for self-training. In both cases we verified that the models are trained sufficiently to fully converge through observing the increase of validation loss. Figure 4 shows the validation curve of the baseline model. Note that the model starts to overfit, and we select the model checkpoint at the lowest point. We also varied the learning rate hyperparameter as 0.0002, 0.0005, and 0.001, which produced BLEU score 15.0, 15.6 (reported in the paper), and 15.5 respectively – our baseline model in previous sections obtained the best performance.
+
+![](images/7a6d6c0d2f3cc5244202f60ed24576279f112c5a9c6ce03d28e3facf38384455.jpg)  
+Figure 4: Validation loss v.s. number of update steps, for the baseline model on WMT100K dataset.
+
+B COMPARISON REGARDING SEPARATE TRAINING, JOINT TRAINING, AND FILTERING
+
+In the paper we perform self-training with separate pseudo-training and fine-tuning steps and always use all monolingual data. However, there are other variants such as joint training or iteratively adding confident examples. Here we compare these variants on WMT100K dataset, noisy self-training uses paraphrase as the perturbation function. For joint training, we tune the upsampling ratio of parallel data just as in back-translation (Edunov et al., 2018). We perform noisy self-training for 3 iterations, and for filtering experiments we iteratively use the most confident 2.5M, 3M, and $3 . 8 \mathbf { M }$ monolingual data respectively in these 3 iterations. Table 7 shows that the filtering process helps joint training but still underperforms separate-training methods by over 1.5 BLEU points. Within separate training filtering produces comparable results to using all data. Since separate training with all data is the simplest method and produces the best performance, we stick to this version in the paper.
+
+Table 7: Ablation analysis on WMT100K dataset.   
+
+<table><tr><td>Methods</td><td>BLEU</td></tr><tr><td>baseline</td><td>15.6</td></tr><tr><td>noisy ST (separate training,all data)</td><td>21.8</td></tr><tr><td>noisy ST (separate training, filtering)</td><td>21.6</td></tr><tr><td>noisy ST (joint training,all data)</td><td>18.8</td></tr><tr><td>noisy ST (joint training, filtering)</td><td>20.0</td></tr></table>
+
+# C ADDITIONAL RESULTS ON THE TOY SUM DATASET
+
+We additionally show the error heat maps of the entire data space on the toy sum datasets for the first two iterations. Here the model at pseudo-training step is initialized as the model from last iteration to clearly examine how the decodings change due to injected noise. As shown in Figure 5, for each iteration the pseudo-training step smooths the space and fine-tuning step benefits from it and greatly reduces the errors
+
+![](images/b22e220951af42f1586b41b8a9cd7905b6e3b987a0106d8ef8ab83b3694ee505.jpg)  
+0024681242242283323484042464505254586062468772488246822946   
+(a) baseline
+
+![](images/d12d4c5d60111fbdf243a8a0afb346c75e2d6bd234209349af3e1f79fd955218.jpg)  
+024681012461820224262832453840424645052458605246加24802846880246
+
+(b) noisy ST (PT, iter=1)
+
+![](images/845ce1e77a3b2e292799b4411d25abdb8445b8418524b79bd2575daa7a8b0cfa.jpg)
+
+![](images/b1fbdbc0dc427a84e2259316715686904477a46f5522d668ab9ec0d0d8994290.jpg)  
+002468101246820224262830234334042464850525458602468024802846802468
+
+(d) noisy ST (PT, iter=2)
+
+![](images/369d8293b4654b92b101ce4c5914c527d705e7163689e0466ededb797700c5de.jpg)  
+Figure 5: Error heat maps on the toy sum dataset over the first two iterations. Deeper color represent larger errors.

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+    {
+        "type": "text",
+        "text": "REVISITING SELF-TRAINING FOR NEURAL SEQUENCE GENERATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Junxian He∗ Carnegie Mellon University junxianh@cs.cmu.edu ",
+        "bbox": [
+            183,
+            172,
+            370,
+            215
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jiatao $\\mathbf { G u } ^ { * }$ , Jiajun Shen, Marc’Aurelio Ranzato Facebook AI Research, New York, NY {jgu,jiajunshen,ranzato}@fb.com ",
+        "bbox": [
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+            172,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            252,
+            544,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Self-training is one of the earliest and simplest semi-supervised methods. The key idea is to augment the original labeled dataset with unlabeled data paired with the model’s prediction (i.e. the pseudo-parallel data). While self-training has been extensively studied on classification problems, in complex sequence generation tasks (e.g. machine translation) it is still unclear how self-training works due to the compositionality of the target space. In this work, we first empirically show that selftraining is able to decently improve the supervised baseline on neural sequence generation tasks. Through careful examination of the performance gains, we find that the perturbation on the hidden states (i.e. dropout) is critical for self-training to benefit from the pseudo-parallel data, which acts as a regularizer and forces the model to yield close predictions for similar unlabeled inputs. Such effect helps the model correct some incorrect predictions on unlabeled data. To further encourage this mechanism, we propose to inject noise to the input space, resulting in a “noisy” version of self-training. Empirical study on standard machine translation and text summarization benchmarks shows that noisy self-training is able to effectively utilize unlabeled data and improve the performance of the supervised baseline by a large margin.1 ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks often require large amounts of labeled data to achieve good performance. However, acquiring labels is a costly process, which motivates research on methods that can effectively utilize unlabeled data to improve performance. Towards this goal, semi-supervised learning (Chapelle et al., 2009) methods that take advantage of both labeled and unlabeled data are a natural starting point. In the context of sequence generation problems, semi-supervised approaches have been shown to work well in some cases. For example, back-translation (Sennrich et al., 2015) makes use of the monolingual data on the target side to improve machine translation systems, latent variable models (Kingma et al., 2014) are employed to incorporate unlabeled source data to facilitate sentence compression (Miao & Blunsom, 2016) or code generation (Yin et al., 2018). ",
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+    {
+        "type": "text",
+        "text": "In this work, we revisit a much older and simpler semi-supervised method, self-training (ST, Scudder (1965)), where a base model trained with labeled data acts as a “teacher” to label the unannotated data, which is then used to augment the original small training set. Then, a “student” model is trained with this new training set to yield the final model. Originally designed for classification problems, common wisdom suggests that this method may be effective only when a good fraction of the predictions on unlabeled samples are correct, otherwise mistakes are going to be reinforced (Zhu & Goldberg, 2009). In the field of natural language processing, some early work have successfully applied self-training to word sense disambiguation (Yarowsky, 1995) and parsing (McClosky et al., 2006; Reichart & Rappoport, 2007; Huang & Harper, 2009). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "However, self-training has not been studied extensively when the target output is natural language. This is partially because in language generation applications (e.g. machine translation) hypotheses are often very far from the ground-truth target, especially in low-resource settings. It is natural to ",
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+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Classic Self-training ",
+        "text_level": 1,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1: Train a base model $f _ { \\theta }$ on $L = \\{ \\pmb { x } _ { i } , \\pmb { y } _ { i } \\} _ { i = 1 } ^ { l }$   \n2: repeat   \n3: Apply $f _ { \\theta }$ to the unlabeled instances $U$   \n4: Select a subset $S \\subset \\{ ( x , f _ { \\pmb \\theta } ( \\pmb x ) ) | \\pmb x \\in U \\}$   \n5: Train a new model $f _ { \\theta }$ on $S \\cup L$   \n6: until convergence or maximum iterations are reached ",
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+    },
+    {
+        "type": "text",
+        "text": "ask whether self-training can be useful at all in this case. While Ueffing (2006) and Zhang & Zong (2016) explored self-training in statistical and neural machine translation, only relatively limited gains were reported and, to the best of our knowledge, it is still unclear what makes self-training work. Moreover, Zhang & Zong (2016) did not update the decoder parameters when using pseudo parallel data noting that “synthetic target parts may negatively influence the decoder model of NMT”. ",
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+    {
+        "type": "text",
+        "text": "In this paper, we aim to answer two questions: (1) How does self-training perform in sequence generation tasks like machine translation and text summarization? Are “bad” pseudo targets indeed catastrophic for self-training? (2) If self-training helps improving the baseline, what contributes to its success? What are the important ingredients to make it work? ",
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+        "type": "text",
+        "text": "Towards this end, we first evaluate self-training on a small-scale machine translation task and empirically observe significant performance gains over the supervised baseline (§3.2), then we perform a comprehensive ablation analysis to understand the key factors that contribute to its success (§3.3). We find that the decoding method to generate pseudo targets accounts for part of the improvement, but more importantly, the perturbation of hidden states – dropout (Hinton et al., 2012) – turns out to be a crucial ingredient to prevent self-training from falling into the same local optimum as the base model, and this is responsible for most of the gains. To understand the role of such noise in self-training, we use a toy experiment to analyze how noise effectively propagates labels to nearby inputs, sometimes helping correct incorrect predictions $( \\ S 4 . 1 )$ . Motivated by this analysis, we propose to inject additional noise by perturbing also the input. Comprehensive experiments on machine translation and text summarization tasks demonstrate the effectiveness of noisy self-training. ",
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+    {
+        "type": "text",
+        "text": "2 SELF-TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Formally, in conditional sequence generation tasks like machine translation, we have a parallel dataset $L = \\{ \\pmb { x } _ { i } , \\pmb { y } _ { i } \\} _ { i = 1 } ^ { l }$ and a large unlabeled dataset $\\boldsymbol { U } = \\{ \\mathbf { x } _ { j } \\} _ { j = l + 1 } ^ { l + u }$ , where $| U | > | L |$ in most cases. As shown in Algorithm 1, classic self-training starts from a base model trained with parallel data $L$ , and iteratively applies the current model to obtain predictions on unlabeled instances $U$ , then it incorporates a subset of the pseudo parallel data $S$ to update the current model. ",
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+    {
+        "type": "text",
+        "text": "There are two key factors: (1) Selection of the subset $S$ . $S$ is usually selected based on some confidence scores (e.g. log probability) (Yarowsky, 1995) but it is also possible for $S$ to be the whole pseudo parallel data (Zhu & Goldberg, 2009). (2) Combination of real and pseudo parallel data. A new model is often trained on the two datasets jointly as in back-translation, but this introduces an additional hyper-parameter to weigh the importance of the parallel data relative to the pseudo data (Edunov et al., 2018). Another way is to treat them separately – first we train the model on pseudo parallel data $S$ , and then fine-tune it on real data $L$ . In our preliminary experiments, we find that the separate training strategy with the whole pseudo parallel dataset (i.e. $\\mathbf { \\bar { \\nabla } } S = \\{ ( \\mathbf { { \\boldsymbol { x } } } , f _ { \\theta } ( \\mathbf { { \\boldsymbol { x } } } ) ) | \\mathbf { { \\boldsymbol { x } } } \\in U \\} )$ produces better or equal performance for neural sequence generation while being simpler. Therefore, in the remainder of this paper we use this simpler setting. We include quantitative comparison regarding joint training, separate training, and pseudo-parallel data filtering in Appendix $\\mathbf { B }$ , where separate training is able to match (or surpass) the performance of joint training. ",
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+    {
+        "type": "text",
+        "text": "In self-training, the unsupervised loss $\\mathcal { L } _ { U }$ from unlabeled instances is defined as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/dc9ddb9a816c55aeb5fd587154faab1e0f7ba66739795e9073238d7e2daaf934.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { U } = - \\mathbb { E } _ { { \\pmb x } \\sim p ( { \\pmb x } ) } \\mathbb { E } _ { { \\pmb y } \\sim p _ { \\pmb \\theta ^ { * } } ( { \\pmb y } | { \\pmb x } ) } \\log p _ { \\pmb \\theta } ( { \\pmb y } | { \\pmb x } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $p ( { \\pmb x } )$ is the empirical data distribution approximated with samples from $S$ , $p _ { \\pmb { \\theta } } ( \\pmb { y } | \\pmb { x } )$ is the conditional distribution defined by the model. $\\pmb { \\theta } ^ { * }$ is the parameter from the last iteration (initially it ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/cb74ae6a0c325b5c592ce367ce7606fca92e073e6899961a6f062be999fa118e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>PT</td><td>FT</td></tr><tr><td>baseline</td><td>1</td><td>15.6</td></tr><tr><td>ST (scratch)</td><td>16.8</td><td>17.9</td></tr><tr><td>ST (baseline)</td><td>16.5</td><td>17.5</td></tr></table>",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/da636825754350288176394216a5f9a294d7de4d224c9b3ed76e0b57cc613735.jpg",
+        "image_caption": [
+            "Figure 1: BLEU on WMT100K dataset from the supervised baseline and different self-training variants. We plot the results over 3 iterations. “ST” denotes self-training while “NST” denotes noisy self training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 1: Test tokenized BLEU on WMT100K. Self-training results are from the first iteration. “Scratch” denotes that the system is initialized randomly and trained from scratch, while “baseline” means it is initialized with the baseline model. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "is set as the parameter of the supervised baseline), and fixed within the current iteration. Eq. 1 reveals the connection between self-training and entropy regularization (Grandvalet & Bengio, 2005). In the context of classification, self-training can be understood from the view of entropy regularization (Lee, 2013), which favors a low-density separation between classes, a commonly assumed prior for semi-supervised learning (Chapelle & Zien, 2005). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 A CASE STUDY ON MACHINE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To examine the effectiveness of self-training on neural sequence generation, we start by analyzing a machine translation task. We then perform ablation analysis to understand the contributing factors of the performance gains. ",
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+    {
+        "type": "text",
+        "text": "3.1 SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We work with the standard WMT 2014 English-German dataset consisting of about 3.9 million training sentence pairs after filtering long and imbalanced pairs. Sentences are encoded using 40K byte-pair codes (Sennrich et al., 2016). As a preliminary experiment, we randomly sample 100K sentences from the training set to train the model and use the remaining English sentences as the unlabeled monolingual data. For convenience, we refer to this dataset as WMT100K. Such synthetic setting allows us to have high-quality unlabeled data to verify the performance of self-training. We train with the Base Transformer architecture (Vaswani et al., 2017) and dropout rate at 0.3. Full training and optimization parameters can be found in Appendix A.1. All experiments throughout this paper including the transformer implementation are based on the fairseq toolkit (Ott et al., 2019), and all results are in terms of case-sensitive tokenized BLEU (Papineni et al., 2002). We use beam search decoding (beam size 5) to create the pseudo targets and to report BLEU on test set. ",
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+    {
+        "type": "text",
+        "text": "3.2 OBSERVATIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Figure 1, we use green bars to show the result of applying self-training for three iterations. We include both (1) pseudo-training $( P T )$ : the first step of self-training where we train a new model (from scratch) using only the pseudo parallel data generated by the current model, and (2) finetuning $( F T )$ : the fine-tuned system using real parallel data based on the pretrained model from the PT step. Note that in the fine-tuning step the system is re-initialized from scratch. Surprisingly, we find that the pseudo-training step at the first iteration is able to improve BLEU even if the model is only trained on its own predictions, and fine-tuning further boosts the performance. The test BLEU keeps improving over the first three iterations, until convergence to outperform the initial baseline by 3 BLEU points. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/be82d5f0f37a364784f117f63dad032f26224f1f144d2c853052973bd847f2fb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>PT</td><td>FT</td></tr><tr><td>baseline baseline (w/o dropout)</td><td>1</td><td>15.6 5.2</td></tr><tr><td>ST (beam search,w/ dropout)</td><td>1 16.5</td><td>17.5</td></tr><tr><td>ST (sampling, w/ dropout)</td><td>16.1</td><td>17.0</td></tr><tr><td>ST (beam search,w/o dropout)</td><td>15.8</td><td>16.3</td></tr><tr><td>ST (sampling, w/o dropout)</td><td>15.5</td><td>16.0</td></tr><tr><td>Noisy ST (beam search, w/o dropout)</td><td></td><td>17.9</td></tr><tr><td>Noisy ST (beam search, w/ dropout)</td><td>15.8 16.6</td><td>19.3</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Ablation study on WMT100K data. For ST and noisy ST, we initialize the model with the baseline and results are from one single iteration. Dropout is varied only in the PT step, while dropout is always applied in FT step. Different decoding methods refer to the strategy used to create the pseudo target. At test time we use beam search decoding for all models. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This behaviour is unexpected because no new information seems to be injected during this iterative process – target sentences of the monolingual data are from the base model’s predictions, thus translation errors are likely to remain, if not magnified. This is different from back-translation where new knowledge may originate from an additional backward translation model and real monolingual targets may help the decoder generate more fluent sentences. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "One straightforward hypothesis is that the added pseudo-parallel data might implicitly change the training trajectory towards a (somehow) better local optimum, given that we train a new model from scratch at each iteration. To rule out this hypothesis, we perform an ablation experiment and initialize $\\pmb { \\theta }$ from the last iteration (i.e. $\\pmb { \\theta } ^ { * }$ ). Formally, based on Eq. 1 we have: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c2e4d3e00c8c52c7e3fd5c0625952185ee9e52895d6779e16a29d04f95a47ea1.jpg",
+        "text": "$$\n\\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { U } | _ { \\pmb { \\theta = \\theta } ^ { \\ast } } = - \\mathbb { E } _ { \\pmb { x } \\sim p ( \\pmb { x } ) } \\left[ \\nabla _ { \\pmb { \\theta } } \\mathbb { E } _ { \\pmb { y } \\sim p _ { \\theta ^ { \\ast } } ( \\pmb { y } | \\pmb { x } ) } \\log p _ { \\theta } ( \\pmb { y } | \\pmb { x } ) | _ { \\pmb { \\theta = \\theta } ^ { \\ast } } \\right] = 0 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "because the conditional log likelihood is maximized when $p _ { \\pmb { \\theta } } ( \\pmb { y } | \\pmb { x } )$ matches the underlying data distribution $p _ { \\pmb { \\theta } ^ { \\ast } } ( \\pmb { y } | \\pmb { x } )$ . Therefore, the parameter $\\pmb \\theta$ should not (at least not significantly) change if we initialize it with $\\pmb { \\theta } ^ { * }$ from the last iteration. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Table 1 shows the comparison results of these two initialization schemes at the first iteration. Surprisingly, continuing training from the baseline model also yields an improvement of 1.9 BLEU points, comparable to initializing from random. While stochastic optimization introduces randomness in the training process, it is startling that continuing training gives such a non-trivial improvement. Next, we investigate the underlying reasons for this. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 THE SECRET BEHIND SELF-TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "To understand why continuing training contradicts Eq. 2 and improves translation performance, we examine possible discrepancies between our assumptions and the actual implementation, and formulate two new hypotheses: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "H1. Decoding Strategy. According to this hypothesis, the gains come from the use of beam search for decoding unlabeled data. Since our focus is a sequence generation task, we decode $\\textbf {  { y } }$ with beam search to approximate the expectation in $\\mathbb { E } _ { { \\pmb { y } } \\sim p _ { \\pmb { \\theta } ^ { * } } ( { \\pmb { y } } | { \\pmb x } ) } \\log { \\bar { p } _ { \\pmb { \\theta } } ( { \\pmb { y } } | { \\pmb x } ) }$ , yielding a biased estimate, while sampling decoding would result in an unbiased Monte Carlo estimator. The results in Table 2 demonstrate that the performance drops by 0.5 BLEU when we change the decoding strategy to sampling, which implies that beam search does contribute a bit to the performance gains. This phenomenon makes sense intuitively since beam search tends to generate higher-quality pseudo targets than sampling, and the subsequent cross-entropy training might benefit from implicitly learning the decoding process. However, the decoding strategy hypothesis does not fully explain it, as we still observe a gain of 1.4 BLEU points over the baseline from sampling decoding with dropout. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "H2. Dropout (Hinton et al., 2012). Eq. 1 and Eq. 2 implicitly ignore a (seemingly) small difference between the model used to produce the pseudo targets and the model used for training: at test/decoding time the model does not use dropout while at training time dropout noise is injected in the model hidden states. At training time, the model is forced to produce the same (pseudo) targets given the same set of inputs and the same parameter set but various noisy versions of the hidden states. The conjecture is that the additional expectation over dropout noise renders Eq. 2 false. To verify this, we remove dropout in the pseudo training step2. The results in Table 2 indicate that without dropout the performance of beam search decoding drops by 1.2 BLEU, just 0.7 BLEU higher than the baseline. Moreover, the pseudo-training performance of sampling without dropout is almost the same as the baseline, which finally agrees with our intuitions from Eq. 2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In summary, Table 2 suggests that beam-search decoding contributes only partially to the performance gains, while the implicit perturbation – dropout – accounts for most of it. However, it is still mysterious why such perturbation results in such large performance gains. If dropout is meant to avoid overfitting and fit the target distribution better in the pseudo-training step, why does it bring advantages over the baseline given that the target distribution is from the baseline model itself ? This is the subject of the investigation in the next section. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "4 NOISE IN SELF-TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "4.1 THE ROLE OF NOISE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One hypothesis as to why noise (perturbation) is beneficial for self-training, is that it enforces local smoothness for this task, that is, semantically similar inputs are mapped to the same or similar targets. Since the assumption that similar input should ideally produce similar target largely holds for most tasks in practice, this smoothing effect of pseudo-training step may provide a favorable regularization for the subsequent finetuning step. Unlike standard regularization in supervised training which is local to the real parallel data, self-training smooths the data space covered by the additional and much larger monolingual data. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/3657d7cbbe89c4652f52d121cd143e90972949df2c0e4be23d31134669f46c48.jpg",
+        "table_caption": [
+            "Table 3: Results on the toy sum dataset. For ST and noisy ST, smoothness (↓) and symmetric (↓) results are from the pseudo-training step, while test errors (↓) are from fine-tuning, all at the first iteration. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>smoothness</td><td>symmetric</td><td>error</td></tr><tr><td>baseline</td><td>9.1</td><td>9.8</td><td>7.6</td></tr><tr><td>ST</td><td>8.2</td><td>9.0</td><td>6.2</td></tr><tr><td>noisy ST</td><td>7.3</td><td>8.2</td><td>4.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To verify this hypothesis more easily, we work with the toy task of summing two integers in the range 0 to 99. We concatenate the two integers and view them as a sequence of digits, the sum is also predicted at the digit level, thus this is still a sequence to sequence task. There are 10000 possible data points in the entire space, and we randomly sample 250 instances for training,3 100 for validation, 5000 for test, and 4000 as the unlabeled data. Test errors are computed as the absolute difference between the predicted integer and the ground-truth integer. We use an LSTM model to tackle this task. We perform self-training for one iteration on this toy sum dataset and initialize the model with the base model to rule out differences due to the initialization. Setup details are in Appendix A.1. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For any integer pair $( x _ { 1 } , x _ { 2 } )$ , we measure local smoothness as the standard deviation of the predictions in a $3 \\times 3$ neighborhood of $( x _ { 1 } , x _ { 2 } )$ . These values are averaged over all the 10000 points to obtain the overall smoothness. We compare smoothness between baseline and ST pseudo-training in Table 3. To demonstrate the effect of smoothing on the fine-tuning step, we also report test errors after fine-tuning. We observe that ST pseudo-training attains better smoothness, which helps reducing test errors in the subsequent fine-tuning step. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One natural question is whether we could further improve performance by encouraging even lower smoothness value, although there is a clear trade-off, as a totally smooth model that outputs a constant value is also a bad predictor. One way to decrease smoothness is by increasing the dropout probability in the pseudo-training step, but a large dropout (like 0.5) makes the model too unstable and slow at converging. Therefore, we consider a simple model-agnostic perturbation process – perturbing the input, which we refer to as noisy self-training (noisy ST). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/5d0d65d405c12883f059b336b7d310b3cef8725e1bfc459455866d0394e0bdcb.jpg",
+        "image_caption": [
+            "Figure 2: Two examples of error heat map on the toy sum dataset that shows the effect of smoothness. The left panel of each composition is from the baseline, and the right one is from the pseudo-training step at the first iteration. $x$ and $y$ axes represent the two input integers. Deeper color represent larger errors. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 NOISY SELF-TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "If we perturb the input during the pseudo-training step, then Eq. 1 would be modified to: ",
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec1e54bcef8ad629e835f67454003a9b74995e7d9ebe92150534c7ddb267e5d9.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { U } = - \\mathbb { E } _ { { \\pmb x } ^ { \\prime } \\sim g ( { \\pmb x } ) , { \\pmb x } \\sim p ( { \\pmb x } ) } \\mathbb { E } _ { { \\pmb y } \\sim p _ { \\theta ^ { * } } ( { \\pmb y } | { \\pmb x } ) } \\log p _ { \\theta } ( { \\pmb y } | { \\pmb x } ^ { \\prime } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $g ( { \\pmb x } )$ is a perturbation function. Note that we apply both input perturbation and dropout in the pseudo-training step for noisy ST throughout the paper, but include ablation analysis in $\\ S 4 . 3$ . We first validate noisy ST in the toy sum task. We shuffle the two integers in the input as the perturbation function. Such perturbation is suitable for this task since it would help the model learn the commutative law as well. To check that, we also measure the symmetry of the output space. Specifically, for any point $( x _ { 1 } , x _ { 2 } )$ , we compute $| f ( x _ { 1 } , x _ { 2 } ) - f ( x _ { 2 } , x _ { 1 } ) |$ and average it over all the points. Both smoothness and symmetry values are reported in Table 3. While we do not explicitly perturb the input at nearby integers, the shuffling perturbation greatly improves the smoothness metric as well. Furthermore, predictions are more symmetric and test errors are reduced. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In order to illustrate the effect of smoothness, in Figure 2 we show two examples of error heat map.4 When a point with large error is surrounded by points with small errors, the labels might propagate due to smoothing and its error is likely to become smaller, resulting in a “self-correcting” behaviour, as demonstrated in the left example of Figure 2. However, the prediction of some points might become worse due to the opposite phenomenon too, as shown in the right example of Figure 2. Therefore, the smoothing effect by itself does not guarantee a performance gain in the pseudotraining step, but fine-tuning benefits from it and seems to consistently improve the baseline in all datasets we experiment with. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.3 OBSERVATIONS ON MACHINE TRANSLATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next, we apply noisy self-training to the more realistic WMT100 translation task. We try two different perturbation functions: (1) Synthetic noise as used in unsupervised MT (Lample et al., 2018), where the input tokens are randomly dropped, masked, and shuffled. We use the default noising parameters as in unsupervised MT but study the influence of noise level in $\\ S 5 . 4$ . (2) Paraphrase. We translate the source English sentences to German and translate it back to obtain a paraphrase as the perturbation. Figure 1 shows the results over three iterations. Noisy ST (NST) greatly outperforms the supervised baseline by over 6 BLEU points and normal ST by 3 BLEU points, while synthetic noise does not exhibit much difference from paraphrasing. Since synthetic noise is much simpler and more general, in the remaining experiments we use synthetic noise unless otherwise specified. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Next, we report an ablation analysis of noisy ST when removing dropout at the pseudo-training step in Table 2. Noisy ST without dropout improves the baseline by 2.3 BLEU points and is comparable to normal ST with dropout. When combined together, noisy ST with dropout produces another 1.4 BLEU improvement, indicating that the two perturbations are complementary. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/16aabb0bd7b6f8b569cd06673c36c54ae9d73a673056399b78240ccbffba69c3.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Methods</td><td colspan=\"2\">WMT English-German</td><td colspan=\"3\">FloRes English-Nepali</td></tr><tr><td>100K (+3.8M mono)</td><td>3.9M (+20M mono)</td><td>En-Origin</td><td>Ne-Origin</td><td>Overall</td></tr><tr><td>baseline</td><td>15.6</td><td>28.3</td><td>6.7</td><td>2.3</td><td>4.8</td></tr><tr><td>BT</td><td>20.5</td><td>1</td><td>8.2</td><td>4.5</td><td>6.5</td></tr><tr><td>noisy ST</td><td>21.4</td><td>29.3</td><td>8.9</td><td>3.5</td><td>6.5</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table 4: Results on two machine translation datasets. For WMT100K, we use the remaining 3.8M English and German sentences from training data as unlabeled data for noisy ST and BT, respectively. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "Our experiments below are designed to examine whether the noisy self-training is generally useful across different sequence generation tasks and resource settings. To this end, we conduct experiments on two machine translation datasets and one text summarization dataset to test the effectiveness under both high-resource and low-resource settings. ",
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+    {
+        "type": "text",
+        "text": "5.1 GENERAL SETUP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We run noisy self-training for three iterations or until performance converges. The model is trained from scratch in the pseudo-training step at each iteration since we found this strategy to work slightly better empirically. Full model and training details for all the experiments can be found in Appendix A.1. In some settings, we also include back-translation (BT, Sennrich et al., 2015) as a reference point, since this is probably the most successful semi-supervised learning method for machine translation. However, we want to emphasize that BT is not directly comparable to ST since they use different resources (ST utilizes the unlabeled data on the source side while BT leverages target monolingual data) and use cases. For example, BT is not very effective when we translate English to extremely low-resource languages where there is almost no in-domain target monolingual data available. We follow the practice in (Edunov et al., 2018) to implement BT where we use unrestricted sampling to translate the target data back to the source. Then, we train the real and pseudo parallel data jointly and tune the upsampling ratio of real parallel data. ",
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+    {
+        "type": "text",
+        "text": "5.2 MACHINE TRANSLATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We test the proposed noisy self-training on a high-resource translation benchmark: WMT14 EnglishGerman and a low-resource translation benchmark: FloRes English-Nepali. ",
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+        "type": "text",
+        "text": "• WMT14 English-German: In addition to WMT100K, we also report results with all 3.9M   \ntraining examples. For WMT100K we use the Base Transformer architecture, and the remaining parallel data as the monolingual data. For the full setting, we use the Big Transformer architecture (Vaswani et al., 2017) and randomly sample 20M English sentences from the News Crawl corpus for noisy ST.   \nFloRes English-Nepali: We evaluate noisy self-training on a low-resource machine translation dataset FloRes (Guzman et al., 2019) from English (en) to Nepali (ne), where we have 560K train- ´ ing pairs and a very weak supervised system that attains BLEU smaller than 5 points. For this dataset we have 3.6M Nepali monolingual instances in total (for BT) but 68M English Wikipedia   \nsentences.5 We randomly sample 5M English sentences for noisy ST. We use the same transformer architecture as in (Guzman et al., 2019). ´ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The overall results are shown in Table 4. For almost all cases in both datasets, the noisy ST outperforms the baselines by a large margin $\\mathrm { . 1 \\sim 5 }$ BLEU scores), and we see that noisy ST still improves the baseline even when this is very weak. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Effect of Domain Mismatch. Test sets of the FloRes benchmark were built with mixed originaltranslationese – some sentences are from English sources and some are from Nepali sources. Intuitively, English monolingual data should be more in-domain with English-origin sentences and ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/1d8eb6ece3e93519a4431e7f449e32b42e8026cd6f78f98948fb6423cab21780.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Methods</td><td colspan=\"3\">100K (+3.7M mono)</td><td colspan=\"3\">640K (+3.2M mono)</td><td colspan=\"3\">3.8M(+4M mono)</td></tr><tr><td>R1</td><td>R2</td><td>RL</td><td>R1</td><td>R2</td><td>RL</td><td>R1</td><td>R2</td><td>RL</td></tr><tr><td>MASS (Song et al., 2019)*</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>38.7</td><td>19.7</td><td>36.0</td></tr><tr><td>baseline</td><td>30.4</td><td>12.4</td><td>27.8</td><td>35.8</td><td>17.0</td><td>33.2</td><td>37.9</td><td>19.0</td><td>35.2</td></tr><tr><td>BT</td><td>32.2</td><td>13.8</td><td>29.6</td><td>37.3</td><td>18.4</td><td>34.6</td><td>1</td><td>一</td><td>一</td></tr><tr><td>noisy ST</td><td>34.1</td><td>15.6</td><td>31.4</td><td>36.6</td><td>18.2</td><td>33.9</td><td>38.6</td><td>19.5</td><td>35.9</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Rouge scores on Gigaword datasets. For the 100K setting we use the remaining 3.7M training data as unlabeled instances for noisy ST and BT. In the $3 . 8 \\mathbf { M }$ setting we use 4M unlabeled data for noisy ST. Stared entry $( * )$ denotes that the system uses a much larger dataset for pretraining. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/b08dfd4e8a0d8b7b4ecf148a482647b43eb4bd984377e7fded1224f6a549cbc2.jpg",
+        "image_caption": [
+            "Figure 3: Analysis of noisy self-training on WMT English-German dataset, demonstrating the effect of parallel data size, monolingual data size, and noise level. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Nepali monolingual data should help more for Nepali-origin sentences. To demonstrate this possible domain-mismatch effect, in Table 4 we report BLEU on the two different test sets separately.6 As expected, ST is very effective when the source sentences originate from English. ",
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+    {
+        "type": "text",
+        "text": "Comparison to Back-Translation. Table 4 shows that noisy ST is able to beat BT on WMT100K and on the en-origin test set of FloRes. In contrast, BT is more effective on the ne-origin test set according to BLEU, which is not surprising as the ne-origin test is likely to benefit more from Nepali than English monolingual data. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 TEXT SUMMARIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We further evaluate noisy self-training on the Gigaword summarization dataset (Rush et al., 2015) that has $3 . 8 \\mathbf { M }$ training sentences. We encode the data with 30K byte-pair codes and use the Base Transformer architecture. Similar to the setting of WMT100K, for Gigaword we create two settings where we sample 100K or 640K training examples and use the remaining as unlabeled data to compare with BT. We also consider the setting where all the 3.8M parallel samples are used and we mine in-domain monolingual data by revisiting the original preprocessing procedure7 and using the ${ \\sim } 4 \\mathbf { M }$ samples that Rush et al. (2015) disregarded because they had low-quality targets. We report ROUGE scores (Lin, 2004) in Table 5. Noisy ST consistently outperforms the baseline in all settings, sometimes by a large margin (100K and 640K). It outperforms BT with 100K parallel data but underperforms with 640K parallel data. We conjecture that BT is still effective in this case because the task is still somewhat symmetric as Gigaword mostly contains short sentences and their compressed summaries. Notably, noisy ST in the full setting approaches the performance of state-of-the-art systems which use much larger datasets for pretraining (Song et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "5.4 ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we focus on the WMT English-German dataset to examine the effect of three factors on noisy self-training: the size of the parallel dataset, the size of the monolingual dataset, and the noise level. All the noisy ST results are after the fine-tuning step. ",
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+    },
+    {
+        "type": "text",
+        "text": "Parallel data size. We fix the monolingual data size as 20M from News Crawl dataset, and vary the parallel data size as shown in Figure 3(a). We use a small LSTM model for 10K, Base Transformer for 100K/640K, and Big Transformer for 3.9M.8 Noisy ST is repeated for three iterations. We see that in all cases noisy ST is able to improve upon the baseline, while the performance gain is larger for intermediate value of the size of the parallel dataset, as expected. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Monolingual data size. We fix the parallel data size to 100K samples, and use the rest 3.8M English sentences from the parallel data as monolingual data. We sample from this set 100K, 500K, 1.5M, and $3 . 8 \\mathbf { M }$ sentences. We also include another point that uses 20M monolingual sentences from a subset of News Crawl dataset. We report performance at the first iteration of noisy ST. Figure 3(b) illustrates that the performance keeps improving as the monolingual data size increases, albeit with diminishing returns. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Noise level. We have shown that noisy ST outperforms ST, but intuitively larger noise must not always be better since at some point it may destroy all the information present in the input. We adopt the WMT100K setting with 100K parallel data and 3.8M monolingual data, and set the word blanking probability in the synthetic noise (Lample et al., 2018) to 0.2 (default number), 0.4, 0.6, and 0.8. We also include the baseline ST without any synthetic noise. Figure 3(c) demonstrates that performance is quite sensitive to noise level, and that intermediate values work best. It is still unclear how to select the noise level a priori, besides the usual hyper-parameter search to maximize BLEU on the validation set. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.5 NOISE PROCESS ON PARALLEL DATA ONLY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we justify whether the proposed noisy self-training process would help the supervised baseline alone without the help of any monolingual data. Similar to the training process on the monolingual data, we first train the model on the noisy source data (pseudo-training), and then finetune it on clean parallel data. Different from using monolingual data, there are two variations here in the “pseudo-training” step: we can either train with the fake target predicted by the model as on monolingual data, or train with the real target paired with noisy source. We denote them as “parallel $^ +$ fake target” and “parallel $^ +$ real target” respectively, and report the performance on WMT100K in Table 6. We use the same synthetic noise as used in previous experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When applying the same noise process to parallel data using fake target, the smoothing effect is not significant since it is restricted into the limited parallel data space, producing marginal improvement over the baseline $( + 0 . 4$ BLEU). As a comparison, 100K monolingual data produces $+ 1 . 0$ BLEU and the effect is enhanced when we increase the monolingual data to $3 . 8 \\mathbf { M }$ , which leads to $+ 3 . 7$ BLEU. Interestingly, pairing the noisy source with real target results in much worse performance than the baseline (-4.3 BLEU), which implies that the use of fake target predicted by the model (i.e. distillation) instead of real target is important for the success of noisy self-training, at least in the case where parallel data size is small. Intuitively, the distilled fake target is simpler and relatively easy for the model to fit, but the real target paired with noisy source makes learning even harder than training with real target and real source, which might lead to a bad starting point for fine-tuning. This issue would be particularly severe when the parallel data size is small, in that case the model would have difficulties to fit real target even with clean source. ",
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+    {
+        "type": "text",
+        "text": "6 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            747,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Self-training belongs to a broader class of “pseudo-label” semi-supervised learning approaches. These approaches all learn from pseudo labels assigned to unlabelled data, with different methods on how to assign such labels. For instance, co-training (Blum & Mitchell, 1998) learns models on two independent feature sets of the same data, and assigns confident labels to unlabeled data from one of the models. Co-training reduces modeling bias by taking into account confidence scores from two models. In the same spirit, democratic co-training (Zhou & Goldman, 2004) or tri-training (Zhou & Li, 2005) trains multiple models with different configurations on the same data feature set, and a subset of the models act as teachers for others. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/48335a0dce2a4e0573fc136219abcc4e012ffebba82a89c4bddaa995ec36295c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>PT</td><td>FT</td></tr><tr><td> parallel baseline</td><td>1</td><td>15.6</td></tr><tr><td>noisy ST,100K mono + fake target</td><td>10.2</td><td>16.6</td></tr><tr><td> noisy ST, 3.8M mono + fake target</td><td>16.6</td><td>19.3</td></tr><tr><td>noisy ST,100K parallel +real target noisy ST,10OK parallel + fake target</td><td>6.7 10.4</td><td>11.3 16.0</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Results on WMT100K data. All results are from one single iteration. “Parallel $^ +$ real/fake target” denotes the noise process applied on parallel data but using real/fake target in the “pseudotraining” step. “Mono $^ +$ fake target” is the normal noisy self-training process described in previous sections. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Another line of more recent work perturb the input or feature space of the student’s inputs as data augmentation techniques. Self-training with dropout or noisy self-training can be viewed as an instantiation of this. These approaches have been very successful on classification tasks (Rasmus et al., 2015; Miyato et al., 2017; Laine & Aila, 2017; Miyato et al., 2018; Xie et al., 2019) given that a reasonable amount of predictions of unlabeled data (at least the ones with high confidence) are correct, but their effect on language generation tasks is largely unknown and poorly understood because the pseudo language targets are often very different from the ground-truth labels. Recent work on sequence generation employs auxiliary decoders (Clark et al., 2018) when processing unlabeled data, overall showing rather limited gains. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper we revisit self-training for neural sequence generation, and show that it can be an effective method to improve generalization, particularly when labeled data is scarce. Through a comprehensive ablation analysis and synthetic experiments, we identify that noise injected during self-training plays a critical role for its success due to its smoothing effect. To encourage this behaviour, we explicitly perturb the input to obtain a new variant of self-training, dubbed noisy selftraining. Experiments on machine translation and text summarization demonstrate the effectiveness of this approach in both low and high resource settings. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We want to thank Peng-Jen Chen for helping set up the FloRes experiments, and Michael Auli, Kyunghyun Cho, and Graham Neubig for insightful discussion about this project. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            660,
+            285,
+            674
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A EXPERIMENTS DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            406,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 SETUP DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            142,
+            330,
+            157
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For all experiments, we optimize with Adam (Kingma & Ba, 2014) using $\\beta _ { 1 } = 0 . 9 , \\beta _ { 2 } = 0 . 9 8 , \\epsilon =$ $1 e - 8$ . All implementations are based on fairseq (Ott et al., 2019), and we basically use the same learning rate schedule and label smoothing as in fairseq examples to train the transformers.9 Except for the toy sum dataset which we runs on a single GPU and each batch contains 32 examples, all other experiments are run on 8 GPUs with an effective batch size of 33K tokens. All experiments are validated with loss on the validation set. For self-training or noisy self-training, the pseudo-training takes 300K synchronous updates while the fine-tuning step takes 100K steps. ",
+        "bbox": [
+            173,
+            174,
+            825,
+            271
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We use the downloading and preprocessing scripts in fairseq to obtain the WMT 2014 EnglishGerman dataset,10 which hold out a small fraction of the original training data as the validation set. ",
+        "bbox": [
+            176,
+            279,
+            820,
+            320
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The model architecture for the toy sum dataset is a single-layer LSTM with word embedding size 32, hidden state size 32, and dropout rate 0.3. The model architecture of WMT10K baseline in Figure 3(a) is a single layer LSTM with word embeddings size 256, hidden state size 256, and dropout rate 0.3. ",
+        "bbox": [
+            174,
+            327,
+            825,
+            383
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 JUSTIFICATION OF THE WMT100K BASELINE",
+        "text_level": 1,
+        "bbox": [
+            176,
+            412,
+            537,
+            426
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We provide more details and evidence to show that our baseline model on WMT100K dataset is trained properly. In all the experiments on WMT100K dataset including baseline and self-training ones, we use Adam optimizer with learning rate 0.0005, which is defaulted in fairseq. We do not use early stop during training but select the best model in terms of the validation loss. We train with 30K update steps for the baseline model and (300K pseudo-training $+ \\ 1 0 0 \\mathrm { K }$ fine-tuning) update steps for self-training. In both cases we verified that the models are trained sufficiently to fully converge through observing the increase of validation loss. Figure 4 shows the validation curve of the baseline model. Note that the model starts to overfit, and we select the model checkpoint at the lowest point. We also varied the learning rate hyperparameter as 0.0002, 0.0005, and 0.001, which produced BLEU score 15.0, 15.6 (reported in the paper), and 15.5 respectively – our baseline model in previous sections obtained the best performance. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/7a6d6c0d2f3cc5244202f60ed24576279f112c5a9c6ce03d28e3facf38384455.jpg",
+        "image_caption": [
+            "Figure 4: Validation loss v.s. number of update steps, for the baseline model on WMT100K dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            369,
+            623,
+            625,
+            814
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B COMPARISON REGARDING SEPARATE TRAINING, JOINT TRAINING, AND FILTERING ",
+        "bbox": [
+            171,
+            103,
+            800,
+            135
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In the paper we perform self-training with separate pseudo-training and fine-tuning steps and always use all monolingual data. However, there are other variants such as joint training or iteratively adding confident examples. Here we compare these variants on WMT100K dataset, noisy self-training uses paraphrase as the perturbation function. For joint training, we tune the upsampling ratio of parallel data just as in back-translation (Edunov et al., 2018). We perform noisy self-training for 3 iterations, and for filtering experiments we iteratively use the most confident 2.5M, 3M, and $3 . 8 \\mathbf { M }$ monolingual data respectively in these 3 iterations. Table 7 shows that the filtering process helps joint training but still underperforms separate-training methods by over 1.5 BLEU points. Within separate training filtering produces comparable results to using all data. Since separate training with all data is the simplest method and produces the best performance, we stick to this version in the paper. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/29a7ffb5b4d5697301e6315d3f4edd16f263056b99c790ac8826993407c233d2.jpg",
+        "table_caption": [
+            "Table 7: Ablation analysis on WMT100K dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>BLEU</td></tr><tr><td>baseline</td><td>15.6</td></tr><tr><td>noisy ST (separate training,all data)</td><td>21.8</td></tr><tr><td>noisy ST (separate training, filtering)</td><td>21.6</td></tr><tr><td>noisy ST (joint training,all data)</td><td>18.8</td></tr><tr><td>noisy ST (joint training, filtering)</td><td>20.0</td></tr></table>",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL RESULTS ON THE TOY SUM DATASET ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 13
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+    {
+        "type": "text",
+        "text": "We additionally show the error heat maps of the entire data space on the toy sum datasets for the first two iterations. Here the model at pseudo-training step is initialized as the model from last iteration to clearly examine how the decodings change due to injected noise. As shown in Figure 5, for each iteration the pseudo-training step smooths the space and fine-tuning step benefits from it and greatly reduces the errors ",
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+            "0024681242242283323484042464505254586062468772488246822946 ",
+            "(a) baseline "
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+            "Figure 5: Error heat maps on the toy sum dataset over the first two iterations. Deeper color represent larger errors. "
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parse/train/ZdJQ8KekIyd/ZdJQ8KekIyd.md

@@ -0,0 +1,522 @@
+# Partial Identification of Counterfactual Distributions
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 This paper investigates the problem of bounding counterfactual queries from a   
+2 combination of observational data and qualitative assumptions about the underlying   
+3 data-generating model. These assumptions are usually represented in the form   
+4 of a causal diagram (Pearl, 1995). We show that all counterfactual distributions   
+5 (over finite observed variables) in an arbitrary causal diagram could be generated   
+6 by a special family of structural causal models (SCMs), compatible with the   
+7 same causal diagram, where unobserved (exogenous) variables are discrete, taking   
+8 values in a finite domain. This entails a reduction in which the space where the   
+9 original, arbitrary SCM lives can be mapped to a dual, more well-behaved space   
+10 where the exogenous variables are discrete, and more easily parametrizable. Using   
+11 this reduction, we translate the bounding problem in the original space into an   
+12 equivalent optimization program in the new space. Solving such programs leads to   
+13 optimal bounds over unknown counterfactuals. Finally, we develop effective Monte   
+14 Carlo algorithms to approximate these optimal bounds from a finite number of   
+15 observational data. Our algorithms are validated extensively on synthetic datasets.
+
+# 16 1 Introduction
+
+17 This paper studies the problem of inferring counterfactual queries from the combination of non  
+18 experimental data (e.g., observational studies) and qualitative assumptions about the data-generating   
+19 process. These assumptions are represented in the form of a causal diagram $\pmb { \mathbb { B 2 } }$ , which is a   
+20 directed acyclic graph where arrows indicate the potential existence of functional relationships among   
+21 corresponding variables; some variables are unobserved. This problem arises in diverse fields such   
+22 as artificial intelligence, statistics, cognitive science, economics, and the health and social sciences.   
+23 For example, when investigating the gender discrimination in college admission, one may ask “what   
+24 would the admission outcome be for a female applicant had she been a male?” Such a counterfactual   
+25 query contains conflicting information: in the real world the applicant is female, in the hypothetical   
+26 world she was not. Therefore, it is not immediately clear how to design effective experimental   
+27 procedures for evaluating counterfactuals, let alone how to compute them from observations alone.   
+28 The problem of identifying counterfactual distributions from the combination of data and a causal   
+29 diagram has been studied in the causal inference literature. First, there exist a complete proof system   
+30 for reasoning about counterfactual queries [19]. While such a system, in principle, is sufficient in   
+31 evaluating any identifiable counterfactual expression, it lacks a proof guideline which determines the   
+32 feasibility of such evaluation efficiently. There are algorithms to determine whether a counterfactual   
+33 distribution is inferrable from all possible controlled experiments [41]. There exist also algorithms   
+34 for identifying path-specific effects from experimental data [1] and observational data [42].   
+35 In practice, however, the combination of quantitative knowledge and observed data does not always   
+36 permit one to point-identify the target counterfactual queries. Partial identification methods concern   
+37 with deriving informative bounds over the target counterfactual probability, even when the target   
+38 itself is non-identifiable. Several algorithms have been developed to bound counterfactuals from the   
+39 combination of observational and experimental data [30, 36, 3, 4, 14, 35, 23, 24, 16, 25, 49].   
+40 In this work, we build on the approach introduced by Balke & Pearl in $\mathbb { \left[ 3 \right] }$ , which involves direct   
+41 discretization of the exogenous domains, also referred to as the principal stratification $\mathbb { L } \mathbb { Z } \mathbb { B } \mathbb { 4 } \mathbb { I }$ Con  
+42 sider the causal diagram of Fig. $\begin{array}{c} \end{array} \boxed { 1 \mathrm { a } }$ where $X , Y , Z$ are binary variables in $\{ 0 , 1 \}$ ; $U$ is an unobserved   
+43 variable taking values in an arbitrary continuous domain. $\boxed { 1 3 }$ showed that domains of $U$ could be   
+44 discretized into 16 equivalent classes without changing the original counterfactual distributions and   
+45 the graphical structure in Fig. 1a. For instance, despite it being induced by an arbitrary distribution   
+46 $P ^ { * } ( u )$ over a continuous domain of the exogenous variable $U$ , the observational distribution $P ( x , y | z )$   
+47 must be reproduced by a generative model of the form $\begin{array} { r } { P ( x , y | z ) = \sum _ { u } P ( x | u , z ) P ( y | x , \dot { u } ) P ( \dot { u } ) } \end{array}$   
+48 where $P ( u )$ is a discrete distribution over a finite exogenous domain $\{ 1 , \ldots , 1 6 \}$ .   
+49 Using the finite-state representation of unobserved variables, $\mathbb { H }$ derived tight bounds on treatment   
+50 effects under the condition of noncompliance in Fig. 1a. [11, 21] applied the parsimony of finite-state   
+51 representation in a Bayesian framework, to obtain credible intervals for the posterior distribution of   
+52 causal effects in noncompliance settings. Despite their optimal guarantees, these bounds are only   
+53 applicable to the specific noncompliance setting in Fig. 1a. For the most general cases, a systematic   
+54 procedure for bounding counterfactual queries in arbitrary causal diagrams is still missing.   
+55 Our goal in this paper is to overcome these challenges. We investigate the expressive power of discrete   
+56 structural causal models (SCMs) $\pmb { \mathbb { B 3 } } \|$ where each unobserved variable is drawn from a discrete   
+57 distribution, takes values in a finite set of states. We show that when inferring about counterfactual   
+58 distributions (over finite observed variables) in an arbitrary causal diagram, one could restrict domains   
+59 of unobserved variables to a finite space without loss of generality. This observation allows us to   
+60 develop novel partial identification algorithms to bound unknown counterfactual probabilities from   
+61 the observational data. More specifically, our contributions are as follows. (1) We introduce a   
+62 special family of discrete SCMs, with finite unobserved domains, and show that it could represent   
+63 all categorical counterfactual distributions in an arbitrary causal diagram. (2) Using this result, we   
+64 translate the original partial identification task into equivalent polynomial programs. Solving such   
+65 programs leads to informative bounds over unknown counterfactual probabilities, which are provably   
+66 optimal. (3) We develop an effective Monte Carlo algorithm to approximate optimal counterfactual   
+67 bounds from a finite number of observational data. Finally, our algorithms are validated extensively   
+68 on synthetic datasets. Given space constraints, all proofs are provided in Appendices A and B.
+
+![](images/6f82c575bdbe37b02df34a95b75fa7432880eccb577020331b7cc4e0d9798dca.jpg)  
+Figure 1: DAGs $\mathrm { \textregistered }$ containing a treatment $X$ , an outcome $Y$ , an ancestor $Z$ , and exogenous variables $U$ ; $Z$ in (a) is also referred to as an instrumental variable.
+
+# 69 1.1 Preliminaries
+
+70 We introduce in this section some basic notations and definitions that will be used throughout the   
+71 paper. We use capital letters to denote variables $( X )$ , small letters for their values $( x )$ and $\Omega _ { X }$ for   
+72 their domains. For an arbitrary set $\boldsymbol { X }$ , let $| X |$ be its cardinality. For convenience, we denote by $P ( { \pmb x } )$   
+73 probabilities $P ( \pmb { X } = \pmb { x } )$ ; for an arbitrary subdomain ${ \mathcal { X } } \subseteq \Omega _ { X }$ , $P ( \mathcal X ) \equiv P ( X \in \mathcal X )$ . Finally, the   
+74 indicator function $\mathbb { 1 } _ { X = x }$ returns 1 if an event $\mathbf { \nabla } X = x$ holds true; otherwise $\mathbb { 1 } _ { X = \pmb { x } } = 0$ .   
+75 The basic semantical framework of our analysis rests on structural causal models (SCMs) [33,   
+76 Ch. 7]. An SCM $M$ is a tuple $\langle V , U , F , P \rangle$ where $V$ is a set of endogenous variables and $U$ is   
+77 a set of exogenous variables. $\pmb { F }$ is a set of functions where each $f _ { V } \in F$ decides values of an   
+78 endogenous variable $V \in V$ taking as argument a combination of other variables in the system. That   
+79 is, $v  f _ { V } ( p a _ { V } , u _ { V } ) , P a _ { V } \subseteq V , U _ { V } \subseteq U$ . Exogenous variables $U \in U$ are mutually independent,   
+80 values of which are drawn from the exogenous distribution $P ( \pmb { u } )$ . Naturally, $M$ induces a joint   
+81 distribution $P ( \pmb { v } )$ over endogenous variables $V$ , called the observational distribution. Each SCM   
+82 is associated with a causal diagram $\mathcal { G }$ (e.g., Fig. 1), which is a directed acyclic graph (DAG) where   
+83 solid nodes represent endogenous variables $V$ , empty nodes represent exogenous variables $U$ and   
+84 arrows represent the arguments $P a _ { V } , U _ { V }$ of each function $f _ { V }$ .   
+85 An intervention on an arbitrary subset $X \subseteq V$ , denoted by ${ \mathrm { d o } } ( { \pmb x } )$ , is an operation where values of   
+86 $\boldsymbol { X }$ are set to constants $_ { \textbf { \em x } }$ , regardless of how they are ordinarily determined. For an SCM $M$ , let   
+87 $M _ { x }$ denote a submodel of $M$ induced by intervention ${ \mathrm { d o } } ( { \pmb x } )$ . For any subset $Y \subseteq V$ , the potential   
+88 response $Y _ { x } ( u )$ is defined as the solution of $\mathbf { Y }$ in the submodel $M _ { x }$ given $U = { \pmb u }$ . Drawing values   
+89 of exogenous variables $U$ following the probability measure $P$ induces a counterfactual variable $Y _ { x }$ .   
+90 Specifically, the event $Y _ { x } = y$ (for short, $\scriptstyle { \mathbf { } } _ { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { \Psi } \mathbf \mathbf { } \mathbf { } \mathbf { \Psi \Psi } \mathbf \mathbf { } \mathbf { \Psi } \mathbf \mathbf { } \mathbf { \Psi \Psi \Psi \mathbf } \mathbf { \Psi \Psi \mathbf } \mathbf \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { \Psi \Psi \Psi \mathbf } \mathbf \mathbf  \Psi \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \mathbf \Psi \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf $ ) can be read as $\mathbf { \sigma } ^ { \bullet } \mathbf { Y }$ would be $\textbf {  { y } }$ had $\boldsymbol { X }$ been $\mathbf { \nabla } _ { \mathbf { x } } , \mathbf { \vec { x } }$ . For any   
+91 subsets $Y , \dots , Z , X , \dots , W \subseteq V$ , the distribution over counterfactuals $Y _ { x } , \dots , Z _ { w }$ is defined as:
+
+$$
+P \left( y _ { x } , \ldots , z _ { w } \right) = \int _ { \Omega _ { U } } \mathbb { 1 } _ { Y _ { x } ( \boldsymbol { u } ) = \boldsymbol { y } } \wedge \cdots \wedge \mathbb { 1 } _ { Z _ { w } ( \boldsymbol { u } ) = z } d P ( \boldsymbol { u } ) .
+$$
+
+92 Distributions of the form $P ( \pmb { y _ { x } } )$ is called the interventional distribution; when the treatment set   
+93 $\boldsymbol { X } = \boldsymbol { \emptyset }$ , $P ( \pmb { y } )$ coincides with the observational distribution. Throughout this paper, we assume   
+94 that endogenous variables $V$ are discrete and finite; while exogenous variables $U$ could take any   
+95 (continuous) value. The counterfactual distribution $P \left( y _ { x } , \ldots , z _ { w } \right)$ defined above is thus a categorical   
+96 distribution. For a more detailed survey on SCMs, we refer readers to $\pmb { \mathbb { B 3 } }$ Ch. 7].
+
+# 97 2 Discretization of Structural Causal Models
+
+For a DAG 98 $\mathcal { G }$ with endogenous $V$ and exogenous variables $U$ , let $P ^ { * }$ denote the collection of all 99 counterfactual distributions over variables $V$ . Formally,
+
+$$
+\pmb { P } ^ { * } = \{ P \left( \pmb { y } _ { \pmb { x } } , \ldots , \pmb { z } _ { w } \right) | \forall \pmb { Y } , \ldots , \pmb { Z } , \pmb { X } , \ldots , \pmb { W } \subseteq \pmb { V } \} .
+$$
+
+100 Let $\mathcal { M }$ be the family of all the SCMs compatible with the causal diagram $\mathcal { G }$ , i.e., $\mathcal { M } =$   
+101 $\{ \forall M \mid { \mathcal { G } } _ { M } = { \mathcal { G } } \} ^ { 1 }$ . Counterfactual distributions in $\mathcal { G }$ are defined as the collection $\{ P _ { M } ^ { * } : \forall M \in \mathcal { M } \}$   
+102 that contains all counterfactual probabilities induced by SCMs $M$ in the candidate family $\mathcal { M }$ . In this   
+103 section, we will show that counterfactual distributions in any causal diagram $\mathcal { G }$ could be generated by   
+104 an alternative family of “generic” SCMs compatible with $\mathcal { G }$ , which we will define later.   
+05 Definition 1 (Counterfactual-Equivalence). For a DAG $\mathcal { G }$ , let ${ \mathcal { M } } , { \mathcal { N } }$ be two sets of SCMs compatible   
+106 with $\mathcal { G }$ . $\mathcal { M }$ and $\mathcal { N }$ are said to be counterfactually equivalent (for short, ctf-equivalent) if for any   
+107 $M \in \mathcal { M }$ , there exists an alternative $N \in \mathcal { N }$ such that $P _ { M } ^ { * } = P _ { N } ^ { * }$ , and vice versa.
+
+Our analysis rests on a special family of SCMs where values of each exogenous variable are drawn from a discrete distribution over a finite set of states.
+
+Definition 2. An SCM $M = \langle V , U , F , P \rangle$ is said to be a discrete SCM if
+
+1. Values of every $U \in U$ are drawn from a discrete distribution $P ( u )$ over a domain $\Omega _ { U }$ ; let $\theta _ { u }$ denote the probability $P ( U = u )$ , for any $u \in \Omega _ { U }$ . f   
+2. Values of every $V \in V$ are decided by function $v  f _ { V } ( p a _ { V } , u _ { V } ) \equiv \xi _ { V } ^ { ( p a _ { V } , u _ { V } ) }$ , where for $\forall p a _ { V } , u _ { V } , \xi _ { V } ^ { ( p a _ { V } , u _ { V } ) }$ is a constant in the finite domain $\Omega _ { V }$ .
+
+115 Given a causal diagram $\mathcal { G }$ , our goal is to construct a family of discrete SCMs $\mathcal { N }$ that is counter  
+116 factually equivalent to the original family of SCMs $\mathcal { M }$ . Our construction utilizes a special type of   
+117 clustering of nodes in the diagram, called the confounded component $\lVert \rVert 5 \rVert$ .   
+118 Definition 3. For an DAG $\mathcal { G }$ , a subset $C \subseteq V$ is a c-component if any pair $X , Y \in C$ is connected   
+19 in $\mathcal { G }$ by a bi-directed path of the form $V _ { 1 }  V _ { 2 }  \cdots  V _ { n }$ , $n = 1 , 2 , \ldots$ , where (1) $V _ { 1 } = X$ ,   
+120 $V _ { n } = Y$ ; (2) $\{ V _ { 1 } , \ldots , V _ { n } \} \subseteq V$ ; and (3) each $V _ { i }  V _ { j }$ is a sequence $V _ { i } \left. U _ { k } \right. V _ { j }$ and $U _ { k } \in U$ .
+
+A c-component $C$ in $\mathcal { G }$ is maximal if there exists no other c-component that contains $C$ . We denote by ${ \mathcal { C } } ( { \mathcal { G } } )$ the collection of all maximal c-components in $\mathcal { G }$ . Naturally, c-components in ${ \mathcal { C } } ( { \mathcal { G } } )$ form a partition over endogenous variables $V$ , which, in turn, defines a partition $\{ \cup _ { V \in C } U _ { V } \mid \forall C \in { \mathcal { C } } ( { \mathcal { G } } ) \}$ over exogenous variables $U$ . Therefore, for every $U \in U$ , there must exist a unique c-component in ${ \mathcal { C } } ( { \mathcal { G } } )$ , denoted by $C _ { U }$ , such that $U \in \cup _ { V \in C _ { U } } U _ { V }$ . For example, exogenous variables $U _ { 1 } , U _ { 2 }$ in Fig. 1a corresponds to c-components $C _ { U _ { 1 } } = \{ Z \}$ and $C _ { U _ { 2 } } = \{ X , Y \}$ respectively; while the causal diagram of Fig. 1b only has a single c-component $\{ X , Y , Z \}$ .
+
+128 Theorem 1. For a DAG $\mathcal { G }$ , consider the following conditions2 : (1) $\mathcal { M }$ is the set of all SCMs   
+129 compatible with $\mathcal { G }$ ; (2) $\mathcal { N }$ is the set of all discrete SCMs compatible with $\mathcal { G }$ where for every $U \in U$ ,   
+130 its cardinality $\begin{array} { r } { \left| \Omega _ { U } \right| = \prod _ { V \in { \cal C } _ { U } } | \Omega _ { P a _ { V } } \mapsto \Omega _ { V } | , } \end{array}$ , i.e., the number of functions mapping from $P a _ { V }$ to   
+131 $V$ for every variable $V$ in the $c$ -component $C _ { U }$ . Then, $\mathcal { M }$ and $\mathcal { N }$ are counterfactually equivalent.   
+132 Thm. $^ 1$ establishes the expressive power of discrete SCMs in representing counterfactual distributions   
+133 in a causal diagram $\mathcal { G }$ . It implies that the counterfactual distribution $P \left( y _ { x } , \ldots , z _ { w } \right)$ in any SCM $M$   
+134 could be generated using a generic model as follows, for $\begin{array} { r } { d _ { U } = \prod _ { V \in { \cal { C } } _ { U } } | \Omega _ { P a _ { V } } \mapsto \Omega _ { V } | } \end{array}$ ,
+
+$$
+P \left( y _ { x } , \ldots , z _ { w } \right) = \sum _ { U \in U } \sum _ { u = 1 , \ldots , d _ { U } } \mathbb { 1 } _ { Y _ { x } ( u ) = y } \wedge \cdots \wedge \mathbb { 1 } _ { Z _ { w } ( u ) = z } \prod _ { U \in U } \theta _ { u } .
+$$
+
+135 Among above quantities, $\theta _ { u }$ are parameters of the exogenous distribution $P ( u )$ over a finite domain   
+136 $\{ 1 , \ldots , d _ { U } \}$ . Counterfactual variables ${ \cal Y } _ { x } ( u )$ are recursively defined as follows:
+
+$$
+Y _ { x } ( u ) = \left\{ Y _ { x } ( u ) \ | \ \forall Y \in Y \right\} , \ \mathrm { w h e r e } \ Y _ { x } ( u ) = \left\{ \begin{array} { l l } { \hfill \hfill \hfill \hfill } { x _ { Y } } & { \mathrm { i f } \ Y \in X } \\  \hfill \hfill \xi _ { Y } ^ { ( \{ V _ { x } ( u ) \} \backslash V \in P a _ { Y } \} , u _ { Y } ) } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+137 where $\pmb { x } _ { Y }$ is the value assigned to variable $Y$ in constants $_ { \textbf { \em x } }$ . As an example, consider the causal   
+138 diagram $\mathcal { G }$ described in Fig. $\boxed { 1 6 }$ where $X , Y , Z$ are binary variables in $\{ 0 , 1 \}$ . Since $\mathcal { G }$ has a single c  
+139 component $\{ X , Y , Z \}$ , exogenous variables $U _ { 1 } , U _ { 2 }$ must share the same cardinality $d$ in the proposed   
+140 family of discrete SCMs $\mathcal { N }$ . It follows from Thm. $^ 1$ the counterfactual distribution $P ( z , x _ { z ^ { \prime } } , y _ { x ^ { \prime } } )$ in   
+141 any SCM compatible with $\mathcal { G }$ could be written as follows:
+
+$$
+P ( z , x _ { z ^ { \prime } } , y _ { x ^ { \prime } } ) = \sum _ { u _ { 1 } , u _ { 2 } = 1 } ^ { d } \mathbb { 1 } _ { \xi _ { z } ^ { ( u _ { 1 } ) } = z } \wedge \mathbb { 1 } _ { \xi _ { x } ^ { ( z ^ { \prime } , u _ { 1 } , u _ { 2 } ) } = x } \wedge \mathbb { 1 } _ { \xi _ { Y } ^ { ( x ^ { \prime } , u _ { 2 } ) } = y } \theta _ { u _ { 1 } } \theta _ { u _ { 2 } } ,
+$$
+
+where 142 $\xi _ { Z } ^ { ( u _ { 1 } ) } , \xi _ { X } ^ { ( z , u _ { 1 } , u _ { 2 } ) } , \xi _ { Y } ^ { ( x , u _ { 2 } ) }$ are parameters taking values in $\{ 0 , 1 \}$ ; $\theta _ { u _ { i } }$ , $i = 1 , 2$ , are probabilities 143 of the discrete distribution $P ( u _ { i } )$ over the finite domain $\{ 1 , \ldots , d \}$ . The cardinality $d = | \Omega _ { Z } | \times$ 144 $| \Omega _ { Z } \mapsto \Omega _ { X } | \times | \Omega _ { X } \mapsto \Omega _ { Y } | = 3 2$ . The total cardinalities of domains for $U _ { 1 } , U _ { 2 }$ are thus $2 d = 6 4$ .
+
+145 Comparison with related work One could naïvely apply the discretization procedure in $\mathbb { \left[ 3 \right] }$ and   
+146 obtain a family of discrete SCMs that are sufficient in representing distributions in an causal diagram.   
+147 However, such parametrization is not necessarily complete. To witness, consider again the causal   
+148 diagram in Fig. $\bar { 1 } 6$ with binary $X , Y , Z$ . Applying the discretization in $\mathbb { \left[ 3 \right] }$ leads to a family of discrete   
+149 SCMs compatible with a different diagram in Fig. $\bigstar$ where the cardinality of exogenous variable   
+150 $U$ is equal to $d = 3 2$ (see Appendix $\bar { \bigstar }$ for details). However, this parametrization fails to capture   
+151 some critical constraints over counterfactual distributions since it does not maintain the original   
+152 structure of the causal diagram. For instance, counterfactual variables $Z$ and $Y _ { x }$ in the original   
+153 diagram of Fig. 1b are independent due to independence restrictions $\pmb { \mathbb { B 3 } }$ Ch. 7.3.2]; while $Z$ and   
+154 $Y _ { x }$ in Fig. $\boxed { 1 \mathrm { c } }$ are generally correlated due to the presence of unobserved confounder $U$ . Compared   
+155 with $\pmb { \Vert 3 \Vert }$ , the discretization method in Thm. $^ 1$ captures all constraints over counterfactual distributions   
+156 while requiring only a factor of $| U |$ increase in the cardinality of exogenous domains.   
+157 More recently, $\mathbb { \lVert 1 5 \rVert }$ proved a special case of Thm. $\bigstar$ for interventional distributions in a specific   
+158 class of causal diagrams that satisfy the running intersection property. When there is no direct arrow   
+159 between endogenous variables, $\pmb { \mathbb { B } } 8 \|$ showed that the observational distribution in a diagram could be   
+160 represented using finite-state exogenous variables. Thm. $^ 1$ generalizes these results by showing that,   
+161 for the first time, all counterfactual distributions in an arbitrary causal diagram could be generated   
+162 using discrete exogenous variables taking values from a finite domain, without any loss of generality.
+
+# 2.1 Partial identification of Counterfactual Distributions
+
+To demonstrate the expressive power of discrete SCMs, we investigate the problem of partial identification of counterfactual distributions. For an SCM $M ^ { * } = \langle V , U , F , P \rangle$ , we are interested in evaluating an arbitrary counterfactual probability $P ( \pmb { y } _ { \pmb { x } } , \dots , \pmb { z } _ { \pmb { w } } )$ . The detailed parametrization of $M ^ { * }$ is unknown. Instead, the learner only has access to the causal diagram $\mathcal { G }$ and the observational distribution $P ( \pmb { v } )$ induced by $M ^ { * }$ . Our goal is to derive an informative bound $[ l , r ]$ from the combination of $\mathcal { G }$ and $P ( \pmb { v } )$ that contains the actual counterfactual probability $P ( \pmb { y } _ { \pmb { x } } , \dots , \pmb { z } _ { \pmb { w } } )$ .
+
+170 Let $\mathcal { N }$ denote the family of discrete SCMs defined in Thm. $\bigstar$ which are compatible with the causal   
+171 diagram $\mathcal { G }$ . We derive a bound $[ l , r ]$ over $P ( \pmb { y } _ { \pmb { x } } , \dots , \pmb { z } _ { \pmb { w } } )$ from the observational data $P ( \pmb { v } )$ by solving   
+172 the following optimization problem:
+
+$$
+[ l , r ] = \operatorname* { m i n } / \operatorname* { m a x } \Big \{ P _ { N } ( \pmb { y _ { x } } , \dots , \pmb { z _ { w } } ) \ | \ \forall N \in \mathcal { N } , P _ { N } ( \pmb { v } ) = P ( \pmb { v } ) \Big \}
+$$
+
+173 For instance, consider again the double-bow diagram $\mathcal { G }$ in Fig. 1b. The observational distribution   
+174 $P ( x , y , z )$ in any discrete SCM in $\mathcal { N }$ could be written as:
+
+$$
+P ( x , y , z ) = \sum _ { u _ { 1 } , u _ { 2 } = 1 } ^ { d } \mathbb { 1 } _ { \xi _ { z } ^ { ( u _ { 1 } ) } = z } \wedge \mathbb { 1 } _ { \xi _ { x } ^ { ( z , u _ { 1 } , u _ { 2 } ) } = x } \wedge \mathbb { 1 } _ { \xi _ { Y } ^ { ( x , u _ { 2 } ) } = y } \theta _ { u _ { 1 } } \theta _ { u _ { 2 } } .
+$$
+
+175 One could derive a bound over the counterfactual distribution $P ( z , x _ { z ^ { \prime } } , y _ { x ^ { \prime } } )$ from the observational   
+176 data $P ( x , y , z )$ by solving polynomial programs which optimize the objective Eq. $( 5 )$ over parameters   
+177 $\theta _ { u _ { 1 } } , \theta _ { u _ { 2 } } , \xi _ { Z } ^ { ( u _ { 1 } ) } , \xi _ { X } ^ { ( z , u _ { 1 } , u _ { 2 } ) } , \xi _ { Y } ^ { ( x , u _ { 2 } ) }$ , subject to the observational constraints Eq. $( 7 )$   
+178 As a corollary, it follows immediately from Thm. $^ 1$ that the solution $[ l , r ]$ of the optimization problem   
+179 Eq. $( 6 )$ is guaranteed to be a valid bound over the unknown counterfactual $P ( \pmb { y } _ { x } , \dots , \pmb { z } _ { w } )$ .   
+180 Corollary 1 (Soundness). Given a $D A G { \mathcal { G } }$ and an observational distribution $P ( \pmb { v } )$ , let $\mathcal { M }$ be the set   
+181 of all SCMs compatible with $\mathcal { G }$ and let $\mathcal { M } _ { o } = \{ \forall M \in \mathcal { M } \mid P _ { M } ( \pmb { v } ) = P ( \pmb { v } ) \}$ . For the solution $[ l , r ]$   
+182 of Eq. (6), $P _ { M } ( \pmb { y _ { x } } , \dotsc , \pmb { z _ { w } } ) \in [ l , r ]$ for any SCM $M \in \mathcal { M } _ { o }$ .   
+183 Since the underlying SCM $M ^ { \ast } \in \mathcal { M } _ { o }$ , Corol. $^ 1$ implies that the derived bound $[ l , r ]$ must contain the   
+184 actual counterfactual probability $P ( \pmb { y } _ { x } , \dots , \pmb { z } _ { w } )$ . Our next result shows that such a bound $[ l , r ]$ is   
+185 provably tight, i.e., it cannot be improved without additional assumptions.   
+186 Corollary 2 (Tightness). Given a DAG $\mathcal { G }$ and an observational distribution $P ( \pmb { v } )$ , let $\mathcal { M }$ be the set   
+187 of all SCMs compatible with $\mathcal { G }$ and let $\mathcal { M } _ { o } = \{ \forall M \in \mathcal { M } \mid P _ { M } ( \pmb { v } ) = P ( \pmb { v } ) \} ,$ . For the solution $[ l , r ]$   
+188 of Eq. $( 6 )$ , there exist SCMs $M _ { 1 } , M _ { 2 } \in \mathcal { M } _ { o }$ such that $P _ { M _ { 1 } } ( y _ { x } , \dots , z _ { w } ) = l$ , $P _ { M _ { 2 } } ( y _ { x } , \dots , z _ { w } ) = r$   
+189 Corol. $2$ confirms the tightness of the bound $[ l , r ]$ obtained from Eq. (6). Suppose there exists a valid   
+190 bound $[ l ^ { \prime } , r ^ { \prime } ]$ strictly contained in $[ l , r ]$ . One could construct from Corol. $\bigstar$ an SCM $M$ compatible   
+191 with the causal diagram $\mathcal { G }$ and the observational distribution $P ( \pmb { v } )$ , but its counterfactual probability   
+192 $P ( \pmb { y } _ { x } , \dots , \pmb { z } _ { w } )$ lies outside $[ l ^ { \prime } , r ^ { \prime } ]$ , which is a contradiction.   
+193 The optimization problem of Eq. $( 6 )$ is reducible to equivalent polynomial programs (see Appendix $\mathbf { E } )$ .   
+194 Despite the soundness and tightness of derived bounds, solving such programs may take exponentially   
+195 long in the most general case $\pmb { \mathbb { Z } } 9 \|$ . Our focus here is upon the causal inference aspect of the problem   
+196 and like earlier discussions we do not specify which solvers are used [3, 4]. In some cases of   
+197 interest, effective approximate planning methods for polynomial programs do exist. Investigating   
+198 these methods is an ongoing subject of research [26, 31, 48, 28, 27].
+
+# 99 3 Bayesian Approach for Partial Identification
+
+This section describes an effective algorithm to approximate the optimal counterfactual bound in Eq. $\textcircled { 6 }$ , provided with finite samples $\bar { \pmb { v } } = \left\{ \pmb { v } ^ { ( n ) } \right\} _ { n = 1 } ^ { \bar { N } }$ drawn from the observational distribution $P ( \pmb { v } )$ , and prior distributions over parameters $\theta _ { u }$ and $\xi _ { V } ^ { ( p a _ { V } , u _ { V } ) }$ (possibly uninformative).
+
+203 We first introduce Markov Chain Monte Carlo (MCMC) algorithms that sample the posterior distribu  
+204 tion $P \left( \theta _ { \mathrm { c t f } } \mid \bar { \mathbf { v } } \right)$ over a counterfactual probability $\theta _ { \mathrm { c t f } } = P \left( \pmb { y _ { x } } , \ldots , \pmb { z _ { w } } \right)$ . More specifically, for every   
+205 $V \in V$ , $\forall p a _ { V } , u _ { V }$ , parameters $\xi _ { V } ^ { ( p a _ { V } , u _ { V } ) }$ are drawn uniformly over the finite domain $\Omega _ { V }$ . For every   
+206 $U \in U$ , exogenous probabilities $\theta _ { u }$ are drawn from a generalized Dirichlet distribution $\pmb { \mathbb { I } }$ . We will   
+207 take the view of a stick-breaking construction $\mathbb { \left[ \left. 4 0 \right\| \right. }$ which successively breaks pieces off a unit-length   
+208 stick with size proportional to random draws from a Beta distribution. Parameters $\theta _ { u }$ are proportions   
+209 of each of the pieces relative to its original size. Formally,
+
+$$
+\forall u = 1 , 2 , \ldots , d _ { U } , \qquad \theta _ { u } = \mu _ { u } \prod _ { i = 1 } ^ { u - 1 } ( 1 - \mu _ { i } ) , \qquad \mu _ { u } \sim \mathtt { B e t a } \left( { \alpha _ { U } ^ { ( u ) } } , \beta _ { U } ^ { ( u ) } \right) ,
+$$
+
+![](images/d74052b275b7d98ccaae19f807f008cb034d418e22761d90955771930346c904.jpg)  
+Figure 2: The data-generating process for the observational data $\left\{ X ^ { ( n ) } , Y ^ { ( n ) } , Z ^ { ( n ) } \right\} _ { n = 1 } ^ { N }$ in an SCM associated with the causal diagram in Fig. 1b. For every exogenous variable $U \in U$ , $\ddot { \theta _ { U } } \overset { \cdot } { = } \{ \theta _ { u } \mid \forall u \}$ . For every endogenous variable $V \in V$ , $\xi _ { V } = \Big \{ \xi _ { V } ^ { ( p a _ { V } , u _ { V } ) } \ | \ \forall p a _ { V } , u _ { V } \Big \} .$ .
+
+210 where $\begin{array} { r } { d _ { U } = \prod _ { V \in { \cal { C } } _ { U } } | \Omega _ { P a _ { V } } \mapsto \Omega _ { V } | } \end{array}$ and $\alpha _ { U } ^ { ( u ) } , \beta _ { U } ^ { ( u ) } > 0$ are hyperparameters. Finally, we truncate   
+211 2 this construction by setting $\mu _ { d _ { U } } = 1$ . Note from Eq. $\textcircled { 8 }$ that all parameters $\theta _ { u }$ for $u > d _ { U }$ are equal   
+212 to zero. As an example, Fig. $2$ shows a graphical representation of the data-generating process over   
+213 parameters ✓u and ⇠ (paV ,uV )V associated with SCMs in Fig. 1b, spanning over $N$ observations.   
+214 Gibbs sampling is a well-known MCMC algorithm that allows one to sample posterior distributions.   
+215 For convenience, we introduce the following notations. Let parameters $\pmb { \theta } = \{ \theta _ { u } | \forall U \in U , \forall u \}$   
+216 and ⇠ = n ⇠ (paV ,uV )V | $\bigstar \bigstar = \Big \{ \xi _ { V } ^ { ( p a _ { V } , u _ { V } ) } | \forall V \in V , \forall p a _ { V } , u _ { V } \Big \}$ . The set $\bar { U } = \left\{ U ^ { ( n ) } \right\} _ { n = 1 } ^ { N }$ are exogenous variables   
+217 affecting N observations V¯ =  V (n) N n=1; we use u¯ to represent their realizations. Our blocked   
+218 Gibbs sampler works by iteratively drawing values from the conditional distributions of variables as   
+219 follows $\lVert \hat { 2 2 } \rVert$ . Detailed derivations of complete conditional distributions are shown in Appendix F.   
+220 Sampling $P \left( \bar { \pmb { u } } | \bar { \pmb { v } } , \pmb { \theta } , \pmb { \xi } \right)$ . Exogenous variables $U ^ { ( n ) }$ , $n = 1 , \ldots , N$ , are mutually independent   
+221 given parameters $\theta , \xi$ . We could draw each $\left( U ^ { ( n ) } \mid \theta , \xi , \bar { V } \right)$ corresponding to the $n$ th observation   
+222 independently. The complete conditional for $U ^ { ( n ) }$ is given by
+
+$$
+P \left( \pmb { u } ^ { ( n ) } \mid \pmb { v } ^ { ( n ) } , \pmb { \theta } , \pmb { \xi } \right) \propto \prod _ { V \in \pmb { V } } \mathbb { 1 } _ { \pmb { \xi } _ { V } ^ { \left( p a _ { V } ^ { ( n ) } , \pmb { u } _ { V } ^ { ( n ) } \right) } = \pmb { v } ^ { ( n ) } } \prod _ { U \in \pmb { U } } \theta _ { u } .
+$$
+
+223 Sampling $P \left( \xi , \pmb \theta \mid \bar { \pmb v } , \bar { \pmb u } \right)$ . Parameters $\xi , \theta$ are independent given $\bar { V } , \bar { U }$ . Therefore, we will derive   
+224 complete conditional $\xi , \theta$ separately. Note that in discrete SCMs, the $n$ th observation of variable   
+225 $V \in V$ is decided by $v ^ { ( n ) }  \xi _ { V } ^ { ( p a _ { V } , u _ { V } ) }$ given $p a _ { V } ^ { ( n ) } = p a _ { V }$ , $u _ { V } ^ { ( n ) } = u _ { V }$ . Thus, draw values of each   
+226 $\xi _ { V } ^ { \left( p a _ { V } , u _ { V } \right) } \in \xi$ from the complete conditional defined as:
+
+$$
+P \left( \xi _ { V } ^ { ( p a _ { V } , u _ { V } ) } \mid \bar { v } , \bar { u } \right) = \left\{ \begin{array} { l l } { \mathbb { 1 } _ { \xi _ { V } ^ { ( p a _ { V } , u _ { V } ) } = v ^ { ( i ) } } } & { \mathrm { i f } \exists i , \mathrm { s . t . } p a _ { V } ^ { ( i ) } = p a _ { V } , u _ { V } ^ { ( i ) } = u _ { V } , } \\ { 1 / | \Omega _ { V } | } & { \mathrm { o t h e r w i s e } . } \end{array} \right.
+$$
+
+Let 227 $\begin{array} { r } { n _ { u } = \sum _ { n = 1 } ^ { N } \mathbb { 1 } _ { u ^ { ( n ) } = u } } \end{array}$ records the number of values in $u ^ { ( n ) }$ that equal to $u$ . By the co of the generalized Dirichlet distribution, the complete conditional of $\theta _ { u }$ is given by, for every $\bar { U } \in \bar { U }$ ,
+
+$$
+\forall u = 1 , 2 , \ldots d _ { U } , \quad \theta _ { u } = \mu _ { u } \prod _ { i = 1 } ^ { u - 1 } ( 1 - \mu _ { i } ) , \quad \mu _ { u } \sim \mathtt { B e t a } \left( \alpha _ { U } ^ { ( u ) } + n _ { u } , \beta _ { U } ^ { ( u ) } + \sum _ { k = u + 1 } ^ { d _ { U } } n _ { k } \right) .
+$$
+
+229 Doing so eventually produces values drawn from the posterior distribution over $\left( \theta , \xi , \bar { U } \mid \bar { V } \right)$ . Given   
+230 parameters $\theta , \xi$ , we compute the counterfactual probability $\theta _ { \mathrm { c t f } } = P ( \pmb { y _ { x } } , \dots , \pmb { z _ { w } } )$ following the   
+231 three-step algorithm in $\bar { \big \| } \bar { 3 3 } \big \|$ which consists of abduction, action, and prediction. Thus computing $\theta _ { \mathrm { c t f } }$   
+232 from each draw $\theta , \xi , { \bar { U } }$ eventually gives us the draw from the posterior distribution $P \left( \theta _ { \mathrm { c t f } } \mid \bar { \mathbf { v } } \right)$ .
+
+# 3.1 Collapsed Gibbs Sampling
+
+234 We also describe an alternative sampler that applies to stick-breaking priors with a known Pólya   
+235 urn characterization. Formally, consider stick-breaking priors in Eq. $( 8 )$ with hyperparameters   
+236 $\alpha _ { U } ^ { ( u ) } = \alpha _ { U } / d _ { U }$ and $\beta _ { U } ^ { ( u ) } = ( d _ { U } - u ) \alpha _ { U } / d _ { U }$ for some real $\alpha _ { U } ~ > ~ 0$ . Let $\bar { U } _ { - n }$ denote the set   
+237 difference ${ \bar { U } } \setminus U ^ { ( n ) }$ ; so does $\bar { V } _ { - n } = \bar { V } \setminus V ^ { ( n ) }$ . Our collapsed Gibbs sampler first iteratively draws   
+238 values from the conditional distribution of $\left( U ^ { ( n ) } \mid \bar { U } _ { - n } , \bar { V } \right)$ , $n = 1 , \ldots , N$ , as follows.
+
+Sampling 239 $P \left( \pmb { u } ^ { ( n ) } \mid \bar { \pmb { v } } , \bar { \pmb { u } } _ { - n } \right)$ . At each iteration, draw $U ^ { ( n ) }$ from the conditional given by
+
+$$
+P \left( \boldsymbol { u } ^ { ( n ) } \mid \bar { \boldsymbol { v } } , \bar { \boldsymbol { u } } _ { - n } \right) \propto \prod _ { V \in V } P \left( \boldsymbol { v } ^ { ( n ) } \mid p a _ { V } ^ { ( n ) } , \boldsymbol { u } _ { V } ^ { ( n ) } , \bar { \boldsymbol { v } } _ { - n } , \bar { \boldsymbol { u } } _ { - n } \right) \prod _ { U \in U } P \left( \boldsymbol { u } ^ { ( n ) } \mid \bar { \boldsymbol { v } } _ { - n } , \bar { \boldsymbol { u } } _ { - n } \right) .
+$$
+
+240 Among quantities in the above equation, for every $V \in V$ ,
+
+$$
+P \left( v ^ { ( n ) } \mid p a _ { V } ^ { ( n ) } , u _ { V } ^ { ( n ) } , \bar { v } _ { - n } , \bar { u } _ { - n } \right) = \left\{ \mathbb { 1 } _ { v ^ { ( n ) } = v ^ { ( i ) } } \ \begin{array} { l } { \mathrm { i f } \exists i \neq n , p a _ { V } ^ { ( i ) } = p a _ { V } ^ { ( n ) } , u _ { V } ^ { ( i ) } = u _ { V } ^ { ( n ) } , } \\ { 1 / | \Omega _ { V } | \ \mathrm { o t h e r w i s e } . } \end{array} \right.
+$$
+
+For every 241 $U \in U$ , let $\bar { u } _ { - n }$ be a set of exogenous samples $\left\{ u ^ { ( 1 ) } , \ldots , u ^ { ( n - 1 ) } , u ^ { ( n + 1 ) } , \ldots , u ^ { ( N ) } \right\} .$ . Let 242 $\{ u _ { 1 } ^ { * } , \ldots , u _ { K } ^ { * } \}$ denote $K$ unique values that samples in $\bar { u } _ { - n }$ take on.
+
+$$
+P \left( u ^ { ( n ) } \mid \bar { v } _ { - n } , \bar { u } _ { - n } \right) = \left\{ \begin{array} { l l } { \displaystyle \frac { n _ { k } ^ { * } + \alpha _ { U } / d _ { U } } { \alpha _ { U } + N - 1 } } & { \mathrm { i f ~ } u ^ { ( n ) } = u _ { k } ^ { * } , \mathrm { ~ f o r ~ } k = 1 , \dots , K } \\ { \displaystyle \frac { \alpha _ { U } ( 1 - K / d _ { U } ) } { \alpha _ { U } + N - 1 } } & { \mathrm { i f ~ } u ^ { ( n ) } \notin \{ u _ { 1 } ^ { * } , \dots , u _ { K } ^ { * } \} } \end{array} \right. .
+$$
+
+where 243 $\begin{array} { r } { n _ { k } ^ { * } = \sum _ { i \neq n } \mathbb { 1 } _ { u ^ { ( i ) } = u _ { k } ^ { * } } } \end{array}$ records the number of values in $\boldsymbol { u } ^ { ( i ) } \in \bar { u } _ { - n }$ that are equal to $\boldsymbol { u } _ { k } ^ { * }$
+
+244 Doing so eventually produces exogenous variables drawn from the posterior distribution of $( \bar { U } \mid \bar { V } )$ .   
+245 We then sample parameters from the posterior distribution of $\left( \theta , \xi \mid \bar { U } , \bar { V } \right)$ ; the complete conditional   
+246 $P \left( \xi , \pmb { \theta } \mid \bar { \pmb { v } } , \bar { \pmb { u } } \right)$ are given in Eqs. $( 1 0 )$ and $\textcircled { 1 1 }$ . Finally, computing $\theta _ { \mathrm { c t f } }$ from each sample $\theta , \xi$ gives   
+247 us a draw from the posterior distribution $P \left( \theta _ { \mathrm { c t f } } \mid \bar { v } \right)$ .
+
+When the cardinality $d _ { U }$ of exogenous domains is high, the collapsed Gibbs sampler described here is more computational efficient than the blocked sampler, since it does not iteratively draw parameters $\theta , \xi$ in the high-dimensional space. Instead, the collapsed sampler only draws $\theta , \xi$ once after samples drawn from the distribution of $( \bar { U } \mid \bar { V } )$ converge. On the other hand, when the cardinality $d _ { U }$ is reasonably low, the blocked Gibbs sampler is preferable since it exhibits better convergence $[ [ 2 2 ] ]$ .
+
+# 3.2 Credible Intervals over Counterfactual Probabilities
+
+Given a MCMC sampler, one could bound the counterfactual probability $\theta _ { \mathrm { c t f } }$ by computing credible intervals from the posterior distribution $P \left( \theta _ { \mathrm { c t f } } \mid \bar { \mathbf { v } } \right)$ .
+
+Definition 4. Fix $\alpha \in [ 0 , 1 )$ . A $1 0 0 ( 1 - \alpha ) \%$ credible interval $[ l _ { \alpha } , r _ { \alpha } ]$ for $\theta _ { \mathrm { c t f } }$ is given by
+
+$$
+l _ { \alpha } = \operatorname* { s u p } \left\{ x \mid P \left( \theta _ { \mathrm { c t } } \leq x \mid \bar { v } \right) = \alpha / 2 \right\} , ~ r _ { \alpha } = \operatorname* { i n f } \left\{ x \mid P \left( \theta _ { \mathrm { c t } } \leq x \mid \bar { v } \right) = 1 - \alpha / 2 \right\} .
+$$
+
+257 For a $1 0 0 ( 1 - \alpha ) \%$ credible interval $[ l _ { \alpha } , r _ { \alpha } ]$ , any counterfactual probability $\theta _ { \mathrm { c t f } }$ that is compatible   
+258 with observational data $\bar { \mathbf { \nabla } } \bar { \mathbf { v } }$ lies between the interval $l _ { \alpha }$ and $r _ { \alpha }$ with probability $1 - \alpha$ . Credible   
+259 intervals have been widely applied for computing bounds over counterfactuals provided with finite   
+260 observations $\underline { { \| 2 0 \| } } , \boxed { 4 7 } , \boxed { 3 7 } , \boxed { 8 } , \boxed { 4 6 } $ . As the number of observational data $N$ grows (to infinite), the $1 0 0 \%$   
+261 credible interval $[ l _ { 0 } , r _ { 0 } ]$ eventually converges to the optimal asymptotic bound $[ l , r ]$ in Eq. $( 6 )$ [11].
+
+Let  ✓(t) T be $T$ samples drawn from $P \left( \theta _ { \mathrm { c t f } } \mid \bar { \mathbf { v } } \right)$ . One could compute the $1 0 0 ( 1 - \alpha ) \%$ credible interval for $\bar { \theta _ { \mathrm { c t f } } }$ using the following consistent estimators $\pmb { \| 3 9 \| }$ :
+
+$$
+\hat { l } _ { \alpha } ( T ) = \theta ^ { ( \lceil ( \alpha / 2 ) T \rceil ) } , \hat { r } _ { \alpha } ( T ) = \theta ^ { ( \lceil ( 1 - \alpha / 2 ) T \rceil ) } ,
+$$
+
+where $\theta ^ { ( \lceil ( \alpha / 2 ) T \rceil ) } , \theta ^ { ( \lceil ( 1 - \alpha / 2 ) T \rceil ) }$ are the $\lceil ( \alpha / 2 ) T \rceil$ th smallest and the $\lceil ( 1 - \alpha / 2 ) T \rceil$ th smallest of $\big \{ \theta ^ { ( t ) } \big \} ^ { 3 } .$ Our next results establish non-asymptotic deviation bounds for the empirical estimates of credible intervals defined in Eq. $\textcircled { 1 6 }$ for finite samples.
+
+Lemma 1. Fix 267 $T > 0$ and $\delta \in ( 0 , 1 )$ . Let function $f ( T , \delta ) = \sqrt { 2 T ^ { - 1 } \ln ( 4 / \delta ) }$ . With probability at least 268 $1 - \delta$ , estimators $\hat { l } _ { \alpha } ( T ) , \hat { r } _ { \alpha } ( T )$ for any $\alpha \in [ 0 , 1 )$ is bounded by
+
+$$
+\hat { l } _ { \alpha } ( T ) \in \left[ l _ { \alpha - f ( T , \delta ) } , l _ { \alpha + f ( T , \delta ) } \right] , \qquad \hat { r } _ { \alpha } ( T ) \in \left[ r _ { \alpha + f ( T , \delta ) } , r _ { \alpha - f ( T , \delta ) } \right] .
+$$
+
+3 For any real $\alpha \in \mathbb { R }$ , $\lceil \alpha \rceil$ denotes the smallest integer $n \in \mathbb { Z }$ larger than $\alpha$ , i.e., $\lceil \alpha \rceil = \operatorname* { m i n } \{ n \in \mathbb { Z } \mid n \geq \alpha \}$
+
+269 We summarize our algorithm, CREDIBLEIN  
+270 TERVAL, in Alg. $1 .$ It takes a credible level   
+271 $\alpha$ and tolerance levels $\delta , \epsilon$ as inputs. In par  
+272 ticular, CREDIBLEINTERVAL repeatedly draw   
+273 $T \ge \lceil 2 \epsilon ^ { - 2 } \ln ( 4 / \delta ) \rceil$ samples from $P \left( \dot { \theta _ { \mathrm { c t f } } } \mid \bar { v } \right)$ .   
+274 It then computes estimates $\hat { l } _ { \alpha } ( T ) , \hat { h } _ { \alpha } ( T )$ from   
+275 drawn samples following Eq. $\textcircled { 1 6 }$ and return   
+276 them as the output. It follows immediately from   
+277 Lem. $\bigstar$ that such a procedure efficiently approx  
+278 imates a $1 0 0 ( 1 - \bar { \alpha } ) \%$ credible interval.
+
+# Algorithm 1: CREDIBLEINTERVAL
+
+1: Input: Credible level $\alpha$ , tolerance level $\delta , \epsilon$ .   
+2: Output: An credible interval $[ l _ { \alpha } , h _ { \alpha } ]$ for $\theta _ { \mathrm { c t f } }$   
+3: Let $\dot { T } = \left. \lceil 2 \epsilon ^ { - 2 } \ln ( 4 / \delta ) \right\rceil$ .   
+4: Draw samples $\big \{ \theta ^ { ( 1 ) } , \dots , \theta ^ { ( T ) } \big \}$ from the   
+posterior distribution $P \left( \theta _ { \mathrm { c t f } } \mid \bar { \mathbf { v } } \right)$ .   
+5: Return interval $\big [ \hat { l } _ { \alpha } ( T ) , \hat { r } _ { \alpha } ( T ) \big ]$ (Eq. (16)
+
+Corollary 3. Fix $\delta \in \mathsf { \Gamma } ( 0 , 1 )$ and $\epsilon > 0$ . With probability at least $1 - \delta$ , the interval $[ { \hat { l } } , { \hat { r } } ] =$ CREDIBLEINTERVAL $( \alpha , \delta , \epsilon )$ for any $\alpha \in [ 0 , 1 )$ is bounded by $\hat { l } \in [ l _ { \alpha - \epsilon } , l _ { \alpha + \epsilon } ]$ and $\hat { r } \in [ r _ { \alpha + \epsilon } , r _ { \alpha - \epsilon } ]$ .
+
+Corol. $3$ implies that any counterfactual parameter $\theta _ { \mathrm { c t f } }$ compatible with observational data $\bar { \mathbf { \nabla } } \bar { \mathbf { v } }$ falls between $[ { \hat { l } } , { \hat { r } } ] =$ CREDIBLEINTERVAL $( \alpha , \delta , \epsilon )$ with probability $P \left( \theta _ { \mathrm { c t f } } \in [ \hat { l } , \hat { r } ] \mid \bar { v } \right) \approx 1 - \alpha \pm \epsilon$ . As the tolerance rate $\epsilon  0 , [ \hat { l } , \hat { r } ]$ converges to a $1 0 0 ( 1 - \alpha ) \%$ credible interval with high probability.
+
+# 284 4 Simulations and Experiments
+
+We demonstrate our algorithms on various simulated SCM instances and a real world patient dataset collected from the International Stroke Trial (IST) $\mathbb { m }$ . Overall, we found that simulation results support our findings and the proposed bounding strategy consistently dominates state-of-art algorithms. When target distributions are identifiable (Experiment 1), our bounds collapse to the actual, unknown counterfactual probabilities. For non-identifiable settings, our algorithm obtains sharp asymptotic bounds when closed-form solutions already exist (Experiments 2 & 3); and improves over state-of-art bounds in other more general cases where the optimal strategy is unknown (Experiment 4).
+
+In all experiments, we evaluate our proposed bounding strategy based on credible intervals $( c i )$ . In particular, we draw $4 \times 1 0 ^ { 3 }$ samples from the posterior distribution over the target counterfactual $\mathbf { \bar { \Psi } } ( \theta _ { \mathrm { c t f } } \mid \bar { V } )$ . This allows us to compute $1 0 0 \%$ credible interval over $\theta _ { \mathrm { c t f } }$ within error $\epsilon = 0 . 0 5$ , with probability at least $1 - \delta = 0 . 9 5$ . As the baseline, we also include the actual counterfactual probability $\theta ^ { * }$ . For details on simulation setups and additional experiments, we refer readers to Appendix C.
+
+297 Experiment 1: Frontdoor Graph This experiment evaluates our sam  
+298 pling algorithm on interventional probabilities that are identifiable from   
+299 the observational data. Consider the “Frontdoor” graph described in   
+300 Fig. $\bigtriangledown$ where $X , Y , W$ are binary variables in $\{ 0 , 1 \}$ ; $U _ { 1 } , U _ { 2 } \in \mathbb { R }$ . In this   
+301 case, the interventional distribution $P ( y _ { x } )$ is identifiable from $P ( x , w , y )$   
+302 through the frontdoor adjustment $\mathbb { \left[ \left. 3 3 \right. \right. }$ Thm. 3.3.4]. We collect $N = 1 0 ^ { 5 }$   
+303 observational samples $\bar { \cal V } = \{ X ^ { ( n ) } , Y ^ { ( n ) } , W ^ { ( n ) } \} _ { n = 1 } ^ { N }$ from a randomly
+
+![](images/51afcc5bc58f94bd670f99dd40c8baf277d6e24ff5e68479eb604bfea3c9b594.jpg)  
+Figure 3: Frontdoor
+
+generated SCM. Fig. 4a shows samples drawn from the posterior distribution of the target probability $\stackrel { \prime } { P } ( Y _ { x = 0 } = 1 ) \mid \bar { V } )$ . The analysis reveals that these samples collapse to the actual interventional probability $P ( Y _ { x = 0 } = 1 ) = 0 . 5 0 8 5$ , which confirms the identifiability of $P ( y _ { x } )$ in Fig. 3.
+
+307 Experiment 2: Instrumental Variables (IV) This experiment evaluates our bounding strategy in   
+308 non-identifiable settings, while closed-form solutions for the optimal bounds over target probabilities   
+309 already exist. Consider first the “IV” diagram in Fig. $^ { 1 \mathrm { a } }$ where $X , Y , Z \in \{ 0 , 1 \}$ and $U _ { 1 } , U _ { 2 } \in \mathbb { R }$   
+310 The non-identifiability of $P ( y _ { x } )$ from the observational data $P ( x , y , z )$ with the instrument $Z$ and the   
+311 unobserved confounding between $X$ and $Y$ has been acknowledged in [5]. For binary $X , Y , Z$ , [2]   
+312 derived closed-form, sharp bounds over $P ( y _ { x } )$ (labelled as opt). We collect $N = 1 0 ^ { 5 }$ observational   
+313 samples $\bar { \cal V } = \{ X ^ { ( n ) } , Y ^ { ( n ) } , Z ^ { ( n ) } \} _ { n = 1 } ^ { N }$ from a randomly generated SCM instance. Fig. $4  { \mathbf { b } }$ shows   
+314 samples drawn from the posterior distribution of  $P ( Y _ { x = 0 } = 1 ) \mid \bar { V } )$ . As a baseline, we also include   
+315 the optimal bound opt, and posterior samples obtained from the Gibbs sampler of $\mathbb { \ m }$ , which utilizes   
+316 the canonical partitions of exogenous domains in $\pmb { \Vert 2 \Vert }$ $( b p )$ . The analysis reveals that our algorithm   
+317 derives the valid bound over the actual probability $P ( Y _ { x = 0 } = 1 ) = 0 . 3 9 5 4$ ; the $1 0 0 \%$ credible   
+318 interval converges to the optimal IV bound $l = 0 . 1 4 6 8 , r = 0 . 6 6 1 7$ .
+
+![](images/974221abcdde98c4222177846c82d05008fed3d1d6f770cc3c9c4ca44560435c.jpg)  
+Figure 4: Histogram plots for samples drawn from the posterior distribution over target counterfactual probabilities. For all plots $( { \bf a } - { \bf d } )$ , ci represents our proposed algorithms; $b p$ stands for Gibbs samplers using the representation of canonical partitions $\pmb { \bigtriangledown } \bar { \bigtriangledown } \bar { \bigtriangledown }$ ; $\theta ^ { * }$ is the actual counterfactual probability. $\mathbf { \underline { { \sigma } } } ( \mathbf { b } , \mathbf { c } )$ opt represents the optimal asymptotic bound, if exists. (d) $n b$ stands for the natural bounds $\pmb { \mathbb { B } } \pmb { \mathbb { O } } \Vert$ .
+
+Experiment 3: Probability of Necessity and Sufficiency (PNS) We now study the problem of evaluating the probability of necessity and sufficiency $P ( Y _ { x = 1 } = 1 , Y _ { x = 0 } = 0 )$ from the observational data $P ( x , y )$ in the “Bow” diagram of Fig. $\boxed { 1 \dot { \mathrm { d } } }$ where $X , Y \in \{ 0 , 1 \}$ and $U \in \mathbb { R }$ . The sharp bound for $P ( Y _ { x = 1 } = 1 , Y _ { x = 0 } = 0 )$ from $P ( x , y )$ was introduced in $\textcircled { \lVert { 4 4 } \rVert }$ (labelled as opt). We collect $N = 1 0 ^ { 5 }$ observational samples $\bar { \cal V } = \{ X ^ { ( n ) } , Y ^ { ( n ) } \} _ { n = 1 } ^ { N }$ from an SCM instance. Fig. $\boxed { 4 \mathrm { c } }$ shows samples drawn from the posterior distribution of $\stackrel { \prime } { P } ( Y _ { x = 1 } = 1 , Y _ { x = 0 } = 0 ) \mid \bar { V } )$ . As a baseline, we also include the optimal bound opt, and posterior samples obtained from the Gibbs sampler which discretizes the exogenous domains using canonical partitions $\mathbb { I } \mathbb { I } \left( b p \right)$ . The analysis reveals that our $1 0 0 \%$ credible interval $( c i )$ matches the optimal PNS bound $l = 0 , r = 0 . 6 7 7 5$ , i.e., the proposed strategy achieves the sharp bound over the counterfactual probability $P ( Y _ { x = 1 } = 1 , Y _ { x = 0 } = 0 ) = 0 . 1 8 6 7$ .
+
+Experiment 4: International Stroke Trials (IST) IST was a large, randomized, open trial of up to 14 days of antithrombotic therapy after stroke onset $\mathbb { \ m }$ . In particular, the treatment $X$ is a pair $( i , j )$ where $i = 0$ stands for no aspirin allocation, 1 otherwise; $j = 0$ stands for no heparin allocation, 1 for median-dosage, and 2 for high-dosage. The primary outcome $Y \in \{ 0 , \ldots , 3 \}$ is the health of the patient 6 months after the treatment, where 0 stands for death, 1 for being dependent on the family, 2 for the partial recovery, and 3 for the full recovery.
+
+To emulate the presence of unobserved confounding, we filter the experimental data with selection rules $f _ { X } ^ { ( Z ) }$ , $Z \in \{ 0 , \ldots , 9 \}$ , following a procedure in $\mathbb { H }$ . Doing so allows us to obtain $N = 3 \times 1 0 ^ { 3 }$ synthetic observational samples $\bar { \cal V } = \{ X ^ { ( n ) } , Y ^ { ( n ) } , Z ^ { ( n ) } \} _ { n = 1 } ^ { N }$ that are compatible with the “Double bow” diagram of Fig. 1b. We are interested in evaluating the treatment effect $E [ Y _ { x = ( 1 , 0 ) } ]$ for only assigning aspirin $\overline { { \boldsymbol { X } } } = ( 1 , 0 )$ . Fig. 4d shows samples drawn from the posterior distribution of $\left( E [ Y _ { x = ( 1 , 0 ) } ] \mid \bar { V } \right)$ . As a baseline, we also include a naïve generalization of the discretization procedure $( b p ) \pmb { \mathbb { Z } }$ (see Appendix ${ \bf D } )$ and the natural bounds $\pm \infty \ldots$ estimated at the $9 5 \%$ confidence level $( n b )$ [49]. Posterior samples of $c i$ and $b p$ are drawn using our proposed collapsed sampler due to the high-dimensional latent space. The analysis reveals that all algorithms achieve bounds that contain the actual, target causal effect $E [ Y _ { x = ( 1 , 0 ) } ] = 1 . 3 4 1 8$ . Our bounding strategy obtains a $1 0 0 \%$ credible interval $l _ { c i } = 1 . 2 6 0 4 , r _ { c i } = 1 . 4 6 8 7$ , which consistently improves over all the other algorithms $( l _ { b p } = 1 . 1 1 2 1$ , $r _ { b p } = 1 . 8 0 7 3 , l _ { n b } = 1 . 1 1 9 5 , r _ { n b } = 1$ $r _ { n b } = 1 . 6 2 2 1$ ).
+
+# 47 5 Conclusion
+
+This paper investigated the problem of partial identification of counterfactual distributions, which concerns with bounding unknown counterfactual probabilities from the combination of the observational data and qualitative assumptions of the data-generating process, represented in the form of a directed acyclic causal diagram. We studied a special family of SCMs with discrete exogenous variables, taking values from a finite set of unobserved states, and showed that it could represent all counterfactual distributions (over finite observed variables) in an arbitrary causal diagram. That is, this new family of discrete SCMs is counterfactual equivalent to the original family of candidate SCMs compatible with the causal diagram. Using this result, we developed a novel algorithm to derive bounds over counterfactual probabilities from finite observations, which are provably tight.
+
+References   
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+[2] A. Balke and J. Pearl. Counterfactual probabilities: Computational methods, bounds, and applications. In R. L. de Mantaras and D. Poole, editors, Uncertainty in Artificial Intelligence 10, pages 46–54. Morgan Kaufmann, San Mateo, CA, 1994.   
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] “Throughout this paper, we assume that endogenous variables $V$ are discrete and finite; while exogenous variables $U$ could take any (continuous) value.”   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] This work does not present any foreseeable societal consequence.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Sec. 1.1. (b) Did you include complete proofs of all theoretical results? [Yes] See Appendices A and B.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See Appendix C.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix C.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix C. “Experiments were performed on a computer with 32GB memory.”
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] “IST was a large, randomized, open trial of up to 14 days of antithrombotic therapy after stroke onset [10].” See also Appendix C   
+(b) Did you mention the license of the assets? [Yes] See Appendix C. The IST dataset is shared under “Open Data Commons Attribution License (ODC-By) v1.0”.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+        "text": "1 This paper investigates the problem of bounding counterfactual queries from a   \n2 combination of observational data and qualitative assumptions about the underlying   \n3 data-generating model. These assumptions are usually represented in the form   \n4 of a causal diagram (Pearl, 1995). We show that all counterfactual distributions   \n5 (over finite observed variables) in an arbitrary causal diagram could be generated   \n6 by a special family of structural causal models (SCMs), compatible with the   \n7 same causal diagram, where unobserved (exogenous) variables are discrete, taking   \n8 values in a finite domain. This entails a reduction in which the space where the   \n9 original, arbitrary SCM lives can be mapped to a dual, more well-behaved space   \n10 where the exogenous variables are discrete, and more easily parametrizable. Using   \n11 this reduction, we translate the bounding problem in the original space into an   \n12 equivalent optimization program in the new space. Solving such programs leads to   \n13 optimal bounds over unknown counterfactuals. Finally, we develop effective Monte   \n14 Carlo algorithms to approximate these optimal bounds from a finite number of   \n15 observational data. Our algorithms are validated extensively on synthetic datasets. ",
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+        "text": "16 1 Introduction ",
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+        "text": "17 This paper studies the problem of inferring counterfactual queries from the combination of non  \n18 experimental data (e.g., observational studies) and qualitative assumptions about the data-generating   \n19 process. These assumptions are represented in the form of a causal diagram $\\pmb { \\mathbb { B 2 } }$ , which is a   \n20 directed acyclic graph where arrows indicate the potential existence of functional relationships among   \n21 corresponding variables; some variables are unobserved. This problem arises in diverse fields such   \n22 as artificial intelligence, statistics, cognitive science, economics, and the health and social sciences.   \n23 For example, when investigating the gender discrimination in college admission, one may ask “what   \n24 would the admission outcome be for a female applicant had she been a male?” Such a counterfactual   \n25 query contains conflicting information: in the real world the applicant is female, in the hypothetical   \n26 world she was not. Therefore, it is not immediately clear how to design effective experimental   \n27 procedures for evaluating counterfactuals, let alone how to compute them from observations alone.   \n28 The problem of identifying counterfactual distributions from the combination of data and a causal   \n29 diagram has been studied in the causal inference literature. First, there exist a complete proof system   \n30 for reasoning about counterfactual queries [19]. While such a system, in principle, is sufficient in   \n31 evaluating any identifiable counterfactual expression, it lacks a proof guideline which determines the   \n32 feasibility of such evaluation efficiently. There are algorithms to determine whether a counterfactual   \n33 distribution is inferrable from all possible controlled experiments [41]. There exist also algorithms   \n34 for identifying path-specific effects from experimental data [1] and observational data [42].   \n35 In practice, however, the combination of quantitative knowledge and observed data does not always   \n36 permit one to point-identify the target counterfactual queries. Partial identification methods concern   \n37 with deriving informative bounds over the target counterfactual probability, even when the target   \n38 itself is non-identifiable. Several algorithms have been developed to bound counterfactuals from the   \n39 combination of observational and experimental data [30, 36, 3, 4, 14, 35, 23, 24, 16, 25, 49].   \n40 In this work, we build on the approach introduced by Balke & Pearl in $\\mathbb { \\left[ 3 \\right] }$ , which involves direct   \n41 discretization of the exogenous domains, also referred to as the principal stratification $\\mathbb { L } \\mathbb { Z } \\mathbb { B } \\mathbb { 4 } \\mathbb { I }$ Con  \n42 sider the causal diagram of Fig. $\\begin{array}{c} \\end{array} \\boxed { 1 \\mathrm { a } }$ where $X , Y , Z$ are binary variables in $\\{ 0 , 1 \\}$ ; $U$ is an unobserved   \n43 variable taking values in an arbitrary continuous domain. $\\boxed { 1 3 }$ showed that domains of $U$ could be   \n44 discretized into 16 equivalent classes without changing the original counterfactual distributions and   \n45 the graphical structure in Fig. 1a. For instance, despite it being induced by an arbitrary distribution   \n46 $P ^ { * } ( u )$ over a continuous domain of the exogenous variable $U$ , the observational distribution $P ( x , y | z )$   \n47 must be reproduced by a generative model of the form $\\begin{array} { r } { P ( x , y | z ) = \\sum _ { u } P ( x | u , z ) P ( y | x , \\dot { u } ) P ( \\dot { u } ) } \\end{array}$   \n48 where $P ( u )$ is a discrete distribution over a finite exogenous domain $\\{ 1 , \\ldots , 1 6 \\}$ .   \n49 Using the finite-state representation of unobserved variables, $\\mathbb { H }$ derived tight bounds on treatment   \n50 effects under the condition of noncompliance in Fig. 1a. [11, 21] applied the parsimony of finite-state   \n51 representation in a Bayesian framework, to obtain credible intervals for the posterior distribution of   \n52 causal effects in noncompliance settings. Despite their optimal guarantees, these bounds are only   \n53 applicable to the specific noncompliance setting in Fig. 1a. For the most general cases, a systematic   \n54 procedure for bounding counterfactual queries in arbitrary causal diagrams is still missing.   \n55 Our goal in this paper is to overcome these challenges. We investigate the expressive power of discrete   \n56 structural causal models (SCMs) $\\pmb { \\mathbb { B 3 } } \\|$ where each unobserved variable is drawn from a discrete   \n57 distribution, takes values in a finite set of states. We show that when inferring about counterfactual   \n58 distributions (over finite observed variables) in an arbitrary causal diagram, one could restrict domains   \n59 of unobserved variables to a finite space without loss of generality. This observation allows us to   \n60 develop novel partial identification algorithms to bound unknown counterfactual probabilities from   \n61 the observational data. More specifically, our contributions are as follows. (1) We introduce a   \n62 special family of discrete SCMs, with finite unobserved domains, and show that it could represent   \n63 all categorical counterfactual distributions in an arbitrary causal diagram. (2) Using this result, we   \n64 translate the original partial identification task into equivalent polynomial programs. Solving such   \n65 programs leads to informative bounds over unknown counterfactual probabilities, which are provably   \n66 optimal. (3) We develop an effective Monte Carlo algorithm to approximate optimal counterfactual   \n67 bounds from a finite number of observational data. Finally, our algorithms are validated extensively   \n68 on synthetic datasets. Given space constraints, all proofs are provided in Appendices A and B. ",
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+            "Figure 1: DAGs $\\mathrm { \\textregistered }$ containing a treatment $X$ , an outcome $Y$ , an ancestor $Z$ , and exogenous variables $U$ ; $Z$ in (a) is also referred to as an instrumental variable. "
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+        "text": "69 1.1 Preliminaries ",
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+        "text": "70 We introduce in this section some basic notations and definitions that will be used throughout the   \n71 paper. We use capital letters to denote variables $( X )$ , small letters for their values $( x )$ and $\\Omega _ { X }$ for   \n72 their domains. For an arbitrary set $\\boldsymbol { X }$ , let $| X |$ be its cardinality. For convenience, we denote by $P ( { \\pmb x } )$   \n73 probabilities $P ( \\pmb { X } = \\pmb { x } )$ ; for an arbitrary subdomain ${ \\mathcal { X } } \\subseteq \\Omega _ { X }$ , $P ( \\mathcal X ) \\equiv P ( X \\in \\mathcal X )$ . Finally, the   \n74 indicator function $\\mathbb { 1 } _ { X = x }$ returns 1 if an event $\\mathbf { \\nabla } X = x$ holds true; otherwise $\\mathbb { 1 } _ { X = \\pmb { x } } = 0$ .   \n75 The basic semantical framework of our analysis rests on structural causal models (SCMs) [33,   \n76 Ch. 7]. An SCM $M$ is a tuple $\\langle V , U , F , P \\rangle$ where $V$ is a set of endogenous variables and $U$ is   \n77 a set of exogenous variables. $\\pmb { F }$ is a set of functions where each $f _ { V } \\in F$ decides values of an   \n78 endogenous variable $V \\in V$ taking as argument a combination of other variables in the system. That   \n79 is, $v  f _ { V } ( p a _ { V } , u _ { V } ) , P a _ { V } \\subseteq V , U _ { V } \\subseteq U$ . Exogenous variables $U \\in U$ are mutually independent,   \n80 values of which are drawn from the exogenous distribution $P ( \\pmb { u } )$ . Naturally, $M$ induces a joint   \n81 distribution $P ( \\pmb { v } )$ over endogenous variables $V$ , called the observational distribution. Each SCM   \n82 is associated with a causal diagram $\\mathcal { G }$ (e.g., Fig. 1), which is a directed acyclic graph (DAG) where   \n83 solid nodes represent endogenous variables $V$ , empty nodes represent exogenous variables $U$ and   \n84 arrows represent the arguments $P a _ { V } , U _ { V }$ of each function $f _ { V }$ .   \n85 An intervention on an arbitrary subset $X \\subseteq V$ , denoted by ${ \\mathrm { d o } } ( { \\pmb x } )$ , is an operation where values of   \n86 $\\boldsymbol { X }$ are set to constants $_ { \\textbf { \\em x } }$ , regardless of how they are ordinarily determined. For an SCM $M$ , let   \n87 $M _ { x }$ denote a submodel of $M$ induced by intervention ${ \\mathrm { d o } } ( { \\pmb x } )$ . For any subset $Y \\subseteq V$ , the potential   \n88 response $Y _ { x } ( u )$ is defined as the solution of $\\mathbf { Y }$ in the submodel $M _ { x }$ given $U = { \\pmb u }$ . Drawing values   \n89 of exogenous variables $U$ following the probability measure $P$ induces a counterfactual variable $Y _ { x }$ .   \n90 Specifically, the event $Y _ { x } = y$ (for short, $\\scriptstyle { \\mathbf { } } _ { \\mathbf { } } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { \\Psi } \\mathbf \\mathbf { } \\mathbf { } \\mathbf { \\Psi \\Psi } \\mathbf \\mathbf { } \\mathbf { \\Psi } \\mathbf \\mathbf { } \\mathbf { \\Psi \\Psi \\Psi \\mathbf } \\mathbf { \\Psi \\Psi \\mathbf } \\mathbf \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { \\Psi \\Psi \\Psi \\mathbf } \\mathbf \\mathbf  \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf $ ) can be read as $\\mathbf { \\sigma } ^ { \\bullet } \\mathbf { Y }$ would be $\\textbf {  { y } }$ had $\\boldsymbol { X }$ been $\\mathbf { \\nabla } _ { \\mathbf { x } } , \\mathbf { \\vec { x } }$ . For any   \n91 subsets $Y , \\dots , Z , X , \\dots , W \\subseteq V$ , the distribution over counterfactuals $Y _ { x } , \\dots , Z _ { w }$ is defined as: ",
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+        "img_path": "images/12802390157662ce27bd6d902483130a0675a2b47b2fe3877517cdc8523ce5ba.jpg",
+        "text": "$$\nP \\left( y _ { x } , \\ldots , z _ { w } \\right) = \\int _ { \\Omega _ { U } } \\mathbb { 1 } _ { Y _ { x } ( \\boldsymbol { u } ) = \\boldsymbol { y } } \\wedge \\cdots \\wedge \\mathbb { 1 } _ { Z _ { w } ( \\boldsymbol { u } ) = z } d P ( \\boldsymbol { u } ) .\n$$",
+        "text_format": "latex",
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+        "text": "92 Distributions of the form $P ( \\pmb { y _ { x } } )$ is called the interventional distribution; when the treatment set   \n93 $\\boldsymbol { X } = \\boldsymbol { \\emptyset }$ , $P ( \\pmb { y } )$ coincides with the observational distribution. Throughout this paper, we assume   \n94 that endogenous variables $V$ are discrete and finite; while exogenous variables $U$ could take any   \n95 (continuous) value. The counterfactual distribution $P \\left( y _ { x } , \\ldots , z _ { w } \\right)$ defined above is thus a categorical   \n96 distribution. For a more detailed survey on SCMs, we refer readers to $\\pmb { \\mathbb { B 3 } }$ Ch. 7]. ",
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+        "type": "text",
+        "text": "97 2 Discretization of Structural Causal Models ",
+        "text_level": 1,
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+        "text": "For a DAG 98 $\\mathcal { G }$ with endogenous $V$ and exogenous variables $U$ , let $P ^ { * }$ denote the collection of all 99 counterfactual distributions over variables $V$ . Formally, ",
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+        "text": "$$\n\\pmb { P } ^ { * } = \\{ P \\left( \\pmb { y } _ { \\pmb { x } } , \\ldots , \\pmb { z } _ { w } \\right) | \\forall \\pmb { Y } , \\ldots , \\pmb { Z } , \\pmb { X } , \\ldots , \\pmb { W } \\subseteq \\pmb { V } \\} .\n$$",
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+        "text": "100 Let $\\mathcal { M }$ be the family of all the SCMs compatible with the causal diagram $\\mathcal { G }$ , i.e., $\\mathcal { M } =$   \n101 $\\{ \\forall M \\mid { \\mathcal { G } } _ { M } = { \\mathcal { G } } \\} ^ { 1 }$ . Counterfactual distributions in $\\mathcal { G }$ are defined as the collection $\\{ P _ { M } ^ { * } : \\forall M \\in \\mathcal { M } \\}$   \n102 that contains all counterfactual probabilities induced by SCMs $M$ in the candidate family $\\mathcal { M }$ . In this   \n103 section, we will show that counterfactual distributions in any causal diagram $\\mathcal { G }$ could be generated by   \n104 an alternative family of “generic” SCMs compatible with $\\mathcal { G }$ , which we will define later.   \n05 Definition 1 (Counterfactual-Equivalence). For a DAG $\\mathcal { G }$ , let ${ \\mathcal { M } } , { \\mathcal { N } }$ be two sets of SCMs compatible   \n106 with $\\mathcal { G }$ . $\\mathcal { M }$ and $\\mathcal { N }$ are said to be counterfactually equivalent (for short, ctf-equivalent) if for any   \n107 $M \\in \\mathcal { M }$ , there exists an alternative $N \\in \\mathcal { N }$ such that $P _ { M } ^ { * } = P _ { N } ^ { * }$ , and vice versa. ",
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+        "type": "text",
+        "text": "Our analysis rests on a special family of SCMs where values of each exogenous variable are drawn from a discrete distribution over a finite set of states. ",
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+        "type": "text",
+        "text": "Definition 2. An SCM $M = \\langle V , U , F , P \\rangle$ is said to be a discrete SCM if ",
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+        "text": "1. Values of every $U \\in U$ are drawn from a discrete distribution $P ( u )$ over a domain $\\Omega _ { U }$ ; let $\\theta _ { u }$ denote the probability $P ( U = u )$ , for any $u \\in \\Omega _ { U }$ . f   \n2. Values of every $V \\in V$ are decided by function $v  f _ { V } ( p a _ { V } , u _ { V } ) \\equiv \\xi _ { V } ^ { ( p a _ { V } , u _ { V } ) }$ , where for $\\forall p a _ { V } , u _ { V } , \\xi _ { V } ^ { ( p a _ { V } , u _ { V } ) }$ is a constant in the finite domain $\\Omega _ { V }$ . ",
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+        "text": "115 Given a causal diagram $\\mathcal { G }$ , our goal is to construct a family of discrete SCMs $\\mathcal { N }$ that is counter  \n116 factually equivalent to the original family of SCMs $\\mathcal { M }$ . Our construction utilizes a special type of   \n117 clustering of nodes in the diagram, called the confounded component $\\lVert \\rVert 5 \\rVert$ .   \n118 Definition 3. For an DAG $\\mathcal { G }$ , a subset $C \\subseteq V$ is a c-component if any pair $X , Y \\in C$ is connected   \n19 in $\\mathcal { G }$ by a bi-directed path of the form $V _ { 1 }  V _ { 2 }  \\cdots  V _ { n }$ , $n = 1 , 2 , \\ldots$ , where (1) $V _ { 1 } = X$ ,   \n120 $V _ { n } = Y$ ; (2) $\\{ V _ { 1 } , \\ldots , V _ { n } \\} \\subseteq V$ ; and (3) each $V _ { i }  V _ { j }$ is a sequence $V _ { i } \\left. U _ { k } \\right. V _ { j }$ and $U _ { k } \\in U$ . ",
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+        "type": "text",
+        "text": "A c-component $C$ in $\\mathcal { G }$ is maximal if there exists no other c-component that contains $C$ . We denote by ${ \\mathcal { C } } ( { \\mathcal { G } } )$ the collection of all maximal c-components in $\\mathcal { G }$ . Naturally, c-components in ${ \\mathcal { C } } ( { \\mathcal { G } } )$ form a partition over endogenous variables $V$ , which, in turn, defines a partition $\\{ \\cup _ { V \\in C } U _ { V } \\mid \\forall C \\in { \\mathcal { C } } ( { \\mathcal { G } } ) \\}$ over exogenous variables $U$ . Therefore, for every $U \\in U$ , there must exist a unique c-component in ${ \\mathcal { C } } ( { \\mathcal { G } } )$ , denoted by $C _ { U }$ , such that $U \\in \\cup _ { V \\in C _ { U } } U _ { V }$ . For example, exogenous variables $U _ { 1 } , U _ { 2 }$ in Fig. 1a corresponds to c-components $C _ { U _ { 1 } } = \\{ Z \\}$ and $C _ { U _ { 2 } } = \\{ X , Y \\}$ respectively; while the causal diagram of Fig. 1b only has a single c-component $\\{ X , Y , Z \\}$ . ",
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+        "text": "128 Theorem 1. For a DAG $\\mathcal { G }$ , consider the following conditions2 : (1) $\\mathcal { M }$ is the set of all SCMs   \n129 compatible with $\\mathcal { G }$ ; (2) $\\mathcal { N }$ is the set of all discrete SCMs compatible with $\\mathcal { G }$ where for every $U \\in U$ ,   \n130 its cardinality $\\begin{array} { r } { \\left| \\Omega _ { U } \\right| = \\prod _ { V \\in { \\cal C } _ { U } } | \\Omega _ { P a _ { V } } \\mapsto \\Omega _ { V } | , } \\end{array}$ , i.e., the number of functions mapping from $P a _ { V }$ to   \n131 $V$ for every variable $V$ in the $c$ -component $C _ { U }$ . Then, $\\mathcal { M }$ and $\\mathcal { N }$ are counterfactually equivalent.   \n132 Thm. $^ 1$ establishes the expressive power of discrete SCMs in representing counterfactual distributions   \n133 in a causal diagram $\\mathcal { G }$ . It implies that the counterfactual distribution $P \\left( y _ { x } , \\ldots , z _ { w } \\right)$ in any SCM $M$   \n134 could be generated using a generic model as follows, for $\\begin{array} { r } { d _ { U } = \\prod _ { V \\in { \\cal { C } } _ { U } } | \\Omega _ { P a _ { V } } \\mapsto \\Omega _ { V } | } \\end{array}$ , ",
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+        "text": "$$\nP \\left( y _ { x } , \\ldots , z _ { w } \\right) = \\sum _ { U \\in U } \\sum _ { u = 1 , \\ldots , d _ { U } } \\mathbb { 1 } _ { Y _ { x } ( u ) = y } \\wedge \\cdots \\wedge \\mathbb { 1 } _ { Z _ { w } ( u ) = z } \\prod _ { U \\in U } \\theta _ { u } .\n$$",
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+        "text": "135 Among above quantities, $\\theta _ { u }$ are parameters of the exogenous distribution $P ( u )$ over a finite domain   \n136 $\\{ 1 , \\ldots , d _ { U } \\}$ . Counterfactual variables ${ \\cal Y } _ { x } ( u )$ are recursively defined as follows: ",
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+        "text": "$$\nY _ { x } ( u ) = \\left\\{ Y _ { x } ( u ) \\ | \\ \\forall Y \\in Y \\right\\} , \\ \\mathrm { w h e r e } \\ Y _ { x } ( u ) = \\left\\{ \\begin{array} { l l } { \\hfill \\hfill \\hfill \\hfill } { x _ { Y } } & { \\mathrm { i f } \\ Y \\in X } \\\\  \\hfill \\hfill \\xi _ { Y } ^ { ( \\{ V _ { x } ( u ) \\} \\backslash V \\in P a _ { Y } \\} , u _ { Y } ) } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+        "text": "137 where $\\pmb { x } _ { Y }$ is the value assigned to variable $Y$ in constants $_ { \\textbf { \\em x } }$ . As an example, consider the causal   \n138 diagram $\\mathcal { G }$ described in Fig. $\\boxed { 1 6 }$ where $X , Y , Z$ are binary variables in $\\{ 0 , 1 \\}$ . Since $\\mathcal { G }$ has a single c  \n139 component $\\{ X , Y , Z \\}$ , exogenous variables $U _ { 1 } , U _ { 2 }$ must share the same cardinality $d$ in the proposed   \n140 family of discrete SCMs $\\mathcal { N }$ . It follows from Thm. $^ 1$ the counterfactual distribution $P ( z , x _ { z ^ { \\prime } } , y _ { x ^ { \\prime } } )$ in   \n141 any SCM compatible with $\\mathcal { G }$ could be written as follows: ",
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+        "img_path": "images/4b7fa0df356c7e0dc79210211d5464c0ad8fdfbe004389fcf97da9d7375d4dac.jpg",
+        "text": "$$\nP ( z , x _ { z ^ { \\prime } } , y _ { x ^ { \\prime } } ) = \\sum _ { u _ { 1 } , u _ { 2 } = 1 } ^ { d } \\mathbb { 1 } _ { \\xi _ { z } ^ { ( u _ { 1 } ) } = z } \\wedge \\mathbb { 1 } _ { \\xi _ { x } ^ { ( z ^ { \\prime } , u _ { 1 } , u _ { 2 } ) } = x } \\wedge \\mathbb { 1 } _ { \\xi _ { Y } ^ { ( x ^ { \\prime } , u _ { 2 } ) } = y } \\theta _ { u _ { 1 } } \\theta _ { u _ { 2 } } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where 142 $\\xi _ { Z } ^ { ( u _ { 1 } ) } , \\xi _ { X } ^ { ( z , u _ { 1 } , u _ { 2 } ) } , \\xi _ { Y } ^ { ( x , u _ { 2 } ) }$ are parameters taking values in $\\{ 0 , 1 \\}$ ; $\\theta _ { u _ { i } }$ , $i = 1 , 2$ , are probabilities 143 of the discrete distribution $P ( u _ { i } )$ over the finite domain $\\{ 1 , \\ldots , d \\}$ . The cardinality $d = | \\Omega _ { Z } | \\times$ 144 $| \\Omega _ { Z } \\mapsto \\Omega _ { X } | \\times | \\Omega _ { X } \\mapsto \\Omega _ { Y } | = 3 2$ . The total cardinalities of domains for $U _ { 1 } , U _ { 2 }$ are thus $2 d = 6 4$ . ",
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+        "text": "145 Comparison with related work One could naïvely apply the discretization procedure in $\\mathbb { \\left[ 3 \\right] }$ and   \n146 obtain a family of discrete SCMs that are sufficient in representing distributions in an causal diagram.   \n147 However, such parametrization is not necessarily complete. To witness, consider again the causal   \n148 diagram in Fig. $\\bar { 1 } 6$ with binary $X , Y , Z$ . Applying the discretization in $\\mathbb { \\left[ 3 \\right] }$ leads to a family of discrete   \n149 SCMs compatible with a different diagram in Fig. $\\bigstar$ where the cardinality of exogenous variable   \n150 $U$ is equal to $d = 3 2$ (see Appendix $\\bar { \\bigstar }$ for details). However, this parametrization fails to capture   \n151 some critical constraints over counterfactual distributions since it does not maintain the original   \n152 structure of the causal diagram. For instance, counterfactual variables $Z$ and $Y _ { x }$ in the original   \n153 diagram of Fig. 1b are independent due to independence restrictions $\\pmb { \\mathbb { B 3 } }$ Ch. 7.3.2]; while $Z$ and   \n154 $Y _ { x }$ in Fig. $\\boxed { 1 \\mathrm { c } }$ are generally correlated due to the presence of unobserved confounder $U$ . Compared   \n155 with $\\pmb { \\Vert 3 \\Vert }$ , the discretization method in Thm. $^ 1$ captures all constraints over counterfactual distributions   \n156 while requiring only a factor of $| U |$ increase in the cardinality of exogenous domains.   \n157 More recently, $\\mathbb { \\lVert 1 5 \\rVert }$ proved a special case of Thm. $\\bigstar$ for interventional distributions in a specific   \n158 class of causal diagrams that satisfy the running intersection property. When there is no direct arrow   \n159 between endogenous variables, $\\pmb { \\mathbb { B } } 8 \\|$ showed that the observational distribution in a diagram could be   \n160 represented using finite-state exogenous variables. Thm. $^ 1$ generalizes these results by showing that,   \n161 for the first time, all counterfactual distributions in an arbitrary causal diagram could be generated   \n162 using discrete exogenous variables taking values from a finite domain, without any loss of generality. ",
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+        "text": "2.1 Partial identification of Counterfactual Distributions ",
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+        "text": "To demonstrate the expressive power of discrete SCMs, we investigate the problem of partial identification of counterfactual distributions. For an SCM $M ^ { * } = \\langle V , U , F , P \\rangle$ , we are interested in evaluating an arbitrary counterfactual probability $P ( \\pmb { y } _ { \\pmb { x } } , \\dots , \\pmb { z } _ { \\pmb { w } } )$ . The detailed parametrization of $M ^ { * }$ is unknown. Instead, the learner only has access to the causal diagram $\\mathcal { G }$ and the observational distribution $P ( \\pmb { v } )$ induced by $M ^ { * }$ . Our goal is to derive an informative bound $[ l , r ]$ from the combination of $\\mathcal { G }$ and $P ( \\pmb { v } )$ that contains the actual counterfactual probability $P ( \\pmb { y } _ { \\pmb { x } } , \\dots , \\pmb { z } _ { \\pmb { w } } )$ . ",
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+        "text": "170 Let $\\mathcal { N }$ denote the family of discrete SCMs defined in Thm. $\\bigstar$ which are compatible with the causal   \n171 diagram $\\mathcal { G }$ . We derive a bound $[ l , r ]$ over $P ( \\pmb { y } _ { \\pmb { x } } , \\dots , \\pmb { z } _ { \\pmb { w } } )$ from the observational data $P ( \\pmb { v } )$ by solving   \n172 the following optimization problem: ",
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+        "text": "$$\n[ l , r ] = \\operatorname* { m i n } / \\operatorname* { m a x } \\Big \\{ P _ { N } ( \\pmb { y _ { x } } , \\dots , \\pmb { z _ { w } } ) \\ | \\ \\forall N \\in \\mathcal { N } , P _ { N } ( \\pmb { v } ) = P ( \\pmb { v } ) \\Big \\}\n$$",
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+        "text": "173 For instance, consider again the double-bow diagram $\\mathcal { G }$ in Fig. 1b. The observational distribution   \n174 $P ( x , y , z )$ in any discrete SCM in $\\mathcal { N }$ could be written as: ",
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+        "text": "$$\nP ( x , y , z ) = \\sum _ { u _ { 1 } , u _ { 2 } = 1 } ^ { d } \\mathbb { 1 } _ { \\xi _ { z } ^ { ( u _ { 1 } ) } = z } \\wedge \\mathbb { 1 } _ { \\xi _ { x } ^ { ( z , u _ { 1 } , u _ { 2 } ) } = x } \\wedge \\mathbb { 1 } _ { \\xi _ { Y } ^ { ( x , u _ { 2 } ) } = y } \\theta _ { u _ { 1 } } \\theta _ { u _ { 2 } } .\n$$",
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+        "text": "175 One could derive a bound over the counterfactual distribution $P ( z , x _ { z ^ { \\prime } } , y _ { x ^ { \\prime } } )$ from the observational   \n176 data $P ( x , y , z )$ by solving polynomial programs which optimize the objective Eq. $( 5 )$ over parameters   \n177 $\\theta _ { u _ { 1 } } , \\theta _ { u _ { 2 } } , \\xi _ { Z } ^ { ( u _ { 1 } ) } , \\xi _ { X } ^ { ( z , u _ { 1 } , u _ { 2 } ) } , \\xi _ { Y } ^ { ( x , u _ { 2 } ) }$ , subject to the observational constraints Eq. $( 7 )$   \n178 As a corollary, it follows immediately from Thm. $^ 1$ that the solution $[ l , r ]$ of the optimization problem   \n179 Eq. $( 6 )$ is guaranteed to be a valid bound over the unknown counterfactual $P ( \\pmb { y } _ { x } , \\dots , \\pmb { z } _ { w } )$ .   \n180 Corollary 1 (Soundness). Given a $D A G { \\mathcal { G } }$ and an observational distribution $P ( \\pmb { v } )$ , let $\\mathcal { M }$ be the set   \n181 of all SCMs compatible with $\\mathcal { G }$ and let $\\mathcal { M } _ { o } = \\{ \\forall M \\in \\mathcal { M } \\mid P _ { M } ( \\pmb { v } ) = P ( \\pmb { v } ) \\}$ . For the solution $[ l , r ]$   \n182 of Eq. (6), $P _ { M } ( \\pmb { y _ { x } } , \\dotsc , \\pmb { z _ { w } } ) \\in [ l , r ]$ for any SCM $M \\in \\mathcal { M } _ { o }$ .   \n183 Since the underlying SCM $M ^ { \\ast } \\in \\mathcal { M } _ { o }$ , Corol. $^ 1$ implies that the derived bound $[ l , r ]$ must contain the   \n184 actual counterfactual probability $P ( \\pmb { y } _ { x } , \\dots , \\pmb { z } _ { w } )$ . Our next result shows that such a bound $[ l , r ]$ is   \n185 provably tight, i.e., it cannot be improved without additional assumptions.   \n186 Corollary 2 (Tightness). Given a DAG $\\mathcal { G }$ and an observational distribution $P ( \\pmb { v } )$ , let $\\mathcal { M }$ be the set   \n187 of all SCMs compatible with $\\mathcal { G }$ and let $\\mathcal { M } _ { o } = \\{ \\forall M \\in \\mathcal { M } \\mid P _ { M } ( \\pmb { v } ) = P ( \\pmb { v } ) \\} ,$ . For the solution $[ l , r ]$   \n188 of Eq. $( 6 )$ , there exist SCMs $M _ { 1 } , M _ { 2 } \\in \\mathcal { M } _ { o }$ such that $P _ { M _ { 1 } } ( y _ { x } , \\dots , z _ { w } ) = l$ , $P _ { M _ { 2 } } ( y _ { x } , \\dots , z _ { w } ) = r$   \n189 Corol. $2$ confirms the tightness of the bound $[ l , r ]$ obtained from Eq. (6). Suppose there exists a valid   \n190 bound $[ l ^ { \\prime } , r ^ { \\prime } ]$ strictly contained in $[ l , r ]$ . One could construct from Corol. $\\bigstar$ an SCM $M$ compatible   \n191 with the causal diagram $\\mathcal { G }$ and the observational distribution $P ( \\pmb { v } )$ , but its counterfactual probability   \n192 $P ( \\pmb { y } _ { x } , \\dots , \\pmb { z } _ { w } )$ lies outside $[ l ^ { \\prime } , r ^ { \\prime } ]$ , which is a contradiction.   \n193 The optimization problem of Eq. $( 6 )$ is reducible to equivalent polynomial programs (see Appendix $\\mathbf { E } )$ .   \n194 Despite the soundness and tightness of derived bounds, solving such programs may take exponentially   \n195 long in the most general case $\\pmb { \\mathbb { Z } } 9 \\|$ . Our focus here is upon the causal inference aspect of the problem   \n196 and like earlier discussions we do not specify which solvers are used [3, 4]. In some cases of   \n197 interest, effective approximate planning methods for polynomial programs do exist. Investigating   \n198 these methods is an ongoing subject of research [26, 31, 48, 28, 27]. ",
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+        "text": "99 3 Bayesian Approach for Partial Identification ",
+        "text_level": 1,
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+        "text": "This section describes an effective algorithm to approximate the optimal counterfactual bound in Eq. $\\textcircled { 6 }$ , provided with finite samples $\\bar { \\pmb { v } } = \\left\\{ \\pmb { v } ^ { ( n ) } \\right\\} _ { n = 1 } ^ { \\bar { N } }$ drawn from the observational distribution $P ( \\pmb { v } )$ , and prior distributions over parameters $\\theta _ { u }$ and $\\xi _ { V } ^ { ( p a _ { V } , u _ { V } ) }$ (possibly uninformative). ",
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+        "text": "203 We first introduce Markov Chain Monte Carlo (MCMC) algorithms that sample the posterior distribu  \n204 tion $P \\left( \\theta _ { \\mathrm { c t f } } \\mid \\bar { \\mathbf { v } } \\right)$ over a counterfactual probability $\\theta _ { \\mathrm { c t f } } = P \\left( \\pmb { y _ { x } } , \\ldots , \\pmb { z _ { w } } \\right)$ . More specifically, for every   \n205 $V \\in V$ , $\\forall p a _ { V } , u _ { V }$ , parameters $\\xi _ { V } ^ { ( p a _ { V } , u _ { V } ) }$ are drawn uniformly over the finite domain $\\Omega _ { V }$ . For every   \n206 $U \\in U$ , exogenous probabilities $\\theta _ { u }$ are drawn from a generalized Dirichlet distribution $\\pmb { \\mathbb { I } }$ . We will   \n207 take the view of a stick-breaking construction $\\mathbb { \\left[ \\left. 4 0 \\right\\| \\right. }$ which successively breaks pieces off a unit-length   \n208 stick with size proportional to random draws from a Beta distribution. Parameters $\\theta _ { u }$ are proportions   \n209 of each of the pieces relative to its original size. Formally, ",
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+        "text": "$$\n\\forall u = 1 , 2 , \\ldots , d _ { U } , \\qquad \\theta _ { u } = \\mu _ { u } \\prod _ { i = 1 } ^ { u - 1 } ( 1 - \\mu _ { i } ) , \\qquad \\mu _ { u } \\sim \\mathtt { B e t a } \\left( { \\alpha _ { U } ^ { ( u ) } } , \\beta _ { U } ^ { ( u ) } \\right) ,\n$$",
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+        "image_caption": [
+            "Figure 2: The data-generating process for the observational data $\\left\\{ X ^ { ( n ) } , Y ^ { ( n ) } , Z ^ { ( n ) } \\right\\} _ { n = 1 } ^ { N }$ in an SCM associated with the causal diagram in Fig. 1b. For every exogenous variable $U \\in U$ , $\\ddot { \\theta _ { U } } \\overset { \\cdot } { = } \\{ \\theta _ { u } \\mid \\forall u \\}$ . For every endogenous variable $V \\in V$ , $\\xi _ { V } = \\Big \\{ \\xi _ { V } ^ { ( p a _ { V } , u _ { V } ) } \\ | \\ \\forall p a _ { V } , u _ { V } \\Big \\} .$ . "
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+        "text": "210 where $\\begin{array} { r } { d _ { U } = \\prod _ { V \\in { \\cal { C } } _ { U } } | \\Omega _ { P a _ { V } } \\mapsto \\Omega _ { V } | } \\end{array}$ and $\\alpha _ { U } ^ { ( u ) } , \\beta _ { U } ^ { ( u ) } > 0$ are hyperparameters. Finally, we truncate   \n211 2 this construction by setting $\\mu _ { d _ { U } } = 1$ . Note from Eq. $\\textcircled { 8 }$ that all parameters $\\theta _ { u }$ for $u > d _ { U }$ are equal   \n212 to zero. As an example, Fig. $2$ shows a graphical representation of the data-generating process over   \n213 parameters ✓u and ⇠ (paV ,uV )V associated with SCMs in Fig. 1b, spanning over $N$ observations.   \n214 Gibbs sampling is a well-known MCMC algorithm that allows one to sample posterior distributions.   \n215 For convenience, we introduce the following notations. Let parameters $\\pmb { \\theta } = \\{ \\theta _ { u } | \\forall U \\in U , \\forall u \\}$   \n216 and ⇠ = n ⇠ (paV ,uV )V | $\\bigstar \\bigstar = \\Big \\{ \\xi _ { V } ^ { ( p a _ { V } , u _ { V } ) } | \\forall V \\in V , \\forall p a _ { V } , u _ { V } \\Big \\}$ . The set $\\bar { U } = \\left\\{ U ^ { ( n ) } \\right\\} _ { n = 1 } ^ { N }$ are exogenous variables   \n217 affecting N observations V¯ = \u0000 V (n) N n=1; we use u¯ to represent their realizations. Our blocked   \n218 Gibbs sampler works by iteratively drawing values from the conditional distributions of variables as   \n219 follows $\\lVert \\hat { 2 2 } \\rVert$ . Detailed derivations of complete conditional distributions are shown in Appendix F.   \n220 Sampling $P \\left( \\bar { \\pmb { u } } | \\bar { \\pmb { v } } , \\pmb { \\theta } , \\pmb { \\xi } \\right)$ . Exogenous variables $U ^ { ( n ) }$ , $n = 1 , \\ldots , N$ , are mutually independent   \n221 given parameters $\\theta , \\xi$ . We could draw each $\\left( U ^ { ( n ) } \\mid \\theta , \\xi , \\bar { V } \\right)$ corresponding to the $n$ th observation   \n222 independently. The complete conditional for $U ^ { ( n ) }$ is given by ",
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+        "img_path": "images/ebf234261398f623b7e13bb3978f7c420a7d86219c0b1394adbce00059ef5d88.jpg",
+        "text": "$$\nP \\left( \\pmb { u } ^ { ( n ) } \\mid \\pmb { v } ^ { ( n ) } , \\pmb { \\theta } , \\pmb { \\xi } \\right) \\propto \\prod _ { V \\in \\pmb { V } } \\mathbb { 1 } _ { \\pmb { \\xi } _ { V } ^ { \\left( p a _ { V } ^ { ( n ) } , \\pmb { u } _ { V } ^ { ( n ) } \\right) } = \\pmb { v } ^ { ( n ) } } \\prod _ { U \\in \\pmb { U } } \\theta _ { u } .\n$$",
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+        "text": "223 Sampling $P \\left( \\xi , \\pmb \\theta \\mid \\bar { \\pmb v } , \\bar { \\pmb u } \\right)$ . Parameters $\\xi , \\theta$ are independent given $\\bar { V } , \\bar { U }$ . Therefore, we will derive   \n224 complete conditional $\\xi , \\theta$ separately. Note that in discrete SCMs, the $n$ th observation of variable   \n225 $V \\in V$ is decided by $v ^ { ( n ) }  \\xi _ { V } ^ { ( p a _ { V } , u _ { V } ) }$ given $p a _ { V } ^ { ( n ) } = p a _ { V }$ , $u _ { V } ^ { ( n ) } = u _ { V }$ . Thus, draw values of each   \n226 $\\xi _ { V } ^ { \\left( p a _ { V } , u _ { V } \\right) } \\in \\xi$ from the complete conditional defined as: ",
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+        "text": "$$\nP \\left( \\xi _ { V } ^ { ( p a _ { V } , u _ { V } ) } \\mid \\bar { v } , \\bar { u } \\right) = \\left\\{ \\begin{array} { l l } { \\mathbb { 1 } _ { \\xi _ { V } ^ { ( p a _ { V } , u _ { V } ) } = v ^ { ( i ) } } } & { \\mathrm { i f } \\exists i , \\mathrm { s . t . } p a _ { V } ^ { ( i ) } = p a _ { V } , u _ { V } ^ { ( i ) } = u _ { V } , } \\\\ { 1 / | \\Omega _ { V } | } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.\n$$",
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+        "text": "Let 227 $\\begin{array} { r } { n _ { u } = \\sum _ { n = 1 } ^ { N } \\mathbb { 1 } _ { u ^ { ( n ) } = u } } \\end{array}$ records the number of values in $u ^ { ( n ) }$ that equal to $u$ . By the co of the generalized Dirichlet distribution, the complete conditional of $\\theta _ { u }$ is given by, for every $\\bar { U } \\in \\bar { U }$ , ",
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+        "text": "$$\n\\forall u = 1 , 2 , \\ldots d _ { U } , \\quad \\theta _ { u } = \\mu _ { u } \\prod _ { i = 1 } ^ { u - 1 } ( 1 - \\mu _ { i } ) , \\quad \\mu _ { u } \\sim \\mathtt { B e t a } \\left( \\alpha _ { U } ^ { ( u ) } + n _ { u } , \\beta _ { U } ^ { ( u ) } + \\sum _ { k = u + 1 } ^ { d _ { U } } n _ { k } \\right) .\n$$",
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+        "text": "229 Doing so eventually produces values drawn from the posterior distribution over $\\left( \\theta , \\xi , \\bar { U } \\mid \\bar { V } \\right)$ . Given   \n230 parameters $\\theta , \\xi$ , we compute the counterfactual probability $\\theta _ { \\mathrm { c t f } } = P ( \\pmb { y _ { x } } , \\dots , \\pmb { z _ { w } } )$ following the   \n231 three-step algorithm in $\\bar { \\big \\| } \\bar { 3 3 } \\big \\|$ which consists of abduction, action, and prediction. Thus computing $\\theta _ { \\mathrm { c t f } }$   \n232 from each draw $\\theta , \\xi , { \\bar { U } }$ eventually gives us the draw from the posterior distribution $P \\left( \\theta _ { \\mathrm { c t f } } \\mid \\bar { \\mathbf { v } } \\right)$ . ",
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+        "type": "text",
+        "text": "3.1 Collapsed Gibbs Sampling ",
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+        "text": "234 We also describe an alternative sampler that applies to stick-breaking priors with a known Pólya   \n235 urn characterization. Formally, consider stick-breaking priors in Eq. $( 8 )$ with hyperparameters   \n236 $\\alpha _ { U } ^ { ( u ) } = \\alpha _ { U } / d _ { U }$ and $\\beta _ { U } ^ { ( u ) } = ( d _ { U } - u ) \\alpha _ { U } / d _ { U }$ for some real $\\alpha _ { U } ~ > ~ 0$ . Let $\\bar { U } _ { - n }$ denote the set   \n237 difference ${ \\bar { U } } \\setminus U ^ { ( n ) }$ ; so does $\\bar { V } _ { - n } = \\bar { V } \\setminus V ^ { ( n ) }$ . Our collapsed Gibbs sampler first iteratively draws   \n238 values from the conditional distribution of $\\left( U ^ { ( n ) } \\mid \\bar { U } _ { - n } , \\bar { V } \\right)$ , $n = 1 , \\ldots , N$ , as follows. ",
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+        "text": "Sampling 239 $P \\left( \\pmb { u } ^ { ( n ) } \\mid \\bar { \\pmb { v } } , \\bar { \\pmb { u } } _ { - n } \\right)$ . At each iteration, draw $U ^ { ( n ) }$ from the conditional given by ",
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+        "img_path": "images/251db991da49342362201a3990fbf9670b4244f7b1d07873738ed9ca7a342d61.jpg",
+        "text": "$$\nP \\left( \\boldsymbol { u } ^ { ( n ) } \\mid \\bar { \\boldsymbol { v } } , \\bar { \\boldsymbol { u } } _ { - n } \\right) \\propto \\prod _ { V \\in V } P \\left( \\boldsymbol { v } ^ { ( n ) } \\mid p a _ { V } ^ { ( n ) } , \\boldsymbol { u } _ { V } ^ { ( n ) } , \\bar { \\boldsymbol { v } } _ { - n } , \\bar { \\boldsymbol { u } } _ { - n } \\right) \\prod _ { U \\in U } P \\left( \\boldsymbol { u } ^ { ( n ) } \\mid \\bar { \\boldsymbol { v } } _ { - n } , \\bar { \\boldsymbol { u } } _ { - n } \\right) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "240 Among quantities in the above equation, for every $V \\in V$ , ",
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+        "text": "$$\nP \\left( v ^ { ( n ) } \\mid p a _ { V } ^ { ( n ) } , u _ { V } ^ { ( n ) } , \\bar { v } _ { - n } , \\bar { u } _ { - n } \\right) = \\left\\{ \\mathbb { 1 } _ { v ^ { ( n ) } = v ^ { ( i ) } } \\ \\begin{array} { l } { \\mathrm { i f } \\exists i \\neq n , p a _ { V } ^ { ( i ) } = p a _ { V } ^ { ( n ) } , u _ { V } ^ { ( i ) } = u _ { V } ^ { ( n ) } , } \\\\ { 1 / | \\Omega _ { V } | \\ \\mathrm { o t h e r w i s e } . } \\end{array} \\right.\n$$",
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+        "type": "text",
+        "text": "For every 241 $U \\in U$ , let $\\bar { u } _ { - n }$ be a set of exogenous samples $\\left\\{ u ^ { ( 1 ) } , \\ldots , u ^ { ( n - 1 ) } , u ^ { ( n + 1 ) } , \\ldots , u ^ { ( N ) } \\right\\} .$ . Let 242 $\\{ u _ { 1 } ^ { * } , \\ldots , u _ { K } ^ { * } \\}$ denote $K$ unique values that samples in $\\bar { u } _ { - n }$ take on. ",
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+        "text": "$$\nP \\left( u ^ { ( n ) } \\mid \\bar { v } _ { - n } , \\bar { u } _ { - n } \\right) = \\left\\{ \\begin{array} { l l } { \\displaystyle \\frac { n _ { k } ^ { * } + \\alpha _ { U } / d _ { U } } { \\alpha _ { U } + N - 1 } } & { \\mathrm { i f ~ } u ^ { ( n ) } = u _ { k } ^ { * } , \\mathrm { ~ f o r ~ } k = 1 , \\dots , K } \\\\ { \\displaystyle \\frac { \\alpha _ { U } ( 1 - K / d _ { U } ) } { \\alpha _ { U } + N - 1 } } & { \\mathrm { i f ~ } u ^ { ( n ) } \\notin \\{ u _ { 1 } ^ { * } , \\dots , u _ { K } ^ { * } \\} } \\end{array} \\right. .\n$$",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "where 243 $\\begin{array} { r } { n _ { k } ^ { * } = \\sum _ { i \\neq n } \\mathbb { 1 } _ { u ^ { ( i ) } = u _ { k } ^ { * } } } \\end{array}$ records the number of values in $\\boldsymbol { u } ^ { ( i ) } \\in \\bar { u } _ { - n }$ that are equal to $\\boldsymbol { u } _ { k } ^ { * }$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "244 Doing so eventually produces exogenous variables drawn from the posterior distribution of $( \\bar { U } \\mid \\bar { V } )$ .   \n245 We then sample parameters from the posterior distribution of $\\left( \\theta , \\xi \\mid \\bar { U } , \\bar { V } \\right)$ ; the complete conditional   \n246 $P \\left( \\xi , \\pmb { \\theta } \\mid \\bar { \\pmb { v } } , \\bar { \\pmb { u } } \\right)$ are given in Eqs. $( 1 0 )$ and $\\textcircled { 1 1 }$ . Finally, computing $\\theta _ { \\mathrm { c t f } }$ from each sample $\\theta , \\xi$ gives   \n247 us a draw from the posterior distribution $P \\left( \\theta _ { \\mathrm { c t f } } \\mid \\bar { v } \\right)$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "When the cardinality $d _ { U }$ of exogenous domains is high, the collapsed Gibbs sampler described here is more computational efficient than the blocked sampler, since it does not iteratively draw parameters $\\theta , \\xi$ in the high-dimensional space. Instead, the collapsed sampler only draws $\\theta , \\xi$ once after samples drawn from the distribution of $( \\bar { U } \\mid \\bar { V } )$ converge. On the other hand, when the cardinality $d _ { U }$ is reasonably low, the blocked Gibbs sampler is preferable since it exhibits better convergence $[ [ 2 2 ] ]$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 Credible Intervals over Counterfactual Probabilities ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Given a MCMC sampler, one could bound the counterfactual probability $\\theta _ { \\mathrm { c t f } }$ by computing credible intervals from the posterior distribution $P \\left( \\theta _ { \\mathrm { c t f } } \\mid \\bar { \\mathbf { v } } \\right)$ . ",
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+    {
+        "type": "text",
+        "text": "Definition 4. Fix $\\alpha \\in [ 0 , 1 )$ . A $1 0 0 ( 1 - \\alpha ) \\%$ credible interval $[ l _ { \\alpha } , r _ { \\alpha } ]$ for $\\theta _ { \\mathrm { c t f } }$ is given by ",
+        "bbox": [
+            173,
+            606,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b0b6872edaa87ce239b1386aa24fadb3c3580400d57b55a5e6fabd29af04bdd1.jpg",
+        "text": "$$\nl _ { \\alpha } = \\operatorname* { s u p } \\left\\{ x \\mid P \\left( \\theta _ { \\mathrm { c t } } \\leq x \\mid \\bar { v } \\right) = \\alpha / 2 \\right\\} , ~ r _ { \\alpha } = \\operatorname* { i n f } \\left\\{ x \\mid P \\left( \\theta _ { \\mathrm { c t } } \\leq x \\mid \\bar { v } \\right) = 1 - \\alpha / 2 \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "257 For a $1 0 0 ( 1 - \\alpha ) \\%$ credible interval $[ l _ { \\alpha } , r _ { \\alpha } ]$ , any counterfactual probability $\\theta _ { \\mathrm { c t f } }$ that is compatible   \n258 with observational data $\\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { v } }$ lies between the interval $l _ { \\alpha }$ and $r _ { \\alpha }$ with probability $1 - \\alpha$ . Credible   \n259 intervals have been widely applied for computing bounds over counterfactuals provided with finite   \n260 observations $\\underline { { \\| 2 0 \\| } } , \\boxed { 4 7 } , \\boxed { 3 7 } , \\boxed { 8 } , \\boxed { 4 6 } $ . As the number of observational data $N$ grows (to infinite), the $1 0 0 \\%$   \n261 credible interval $[ l _ { 0 } , r _ { 0 } ]$ eventually converges to the optimal asymptotic bound $[ l , r ]$ in Eq. $( 6 )$ [11]. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Let \u0000 ✓(t) T be $T$ samples drawn from $P \\left( \\theta _ { \\mathrm { c t f } } \\mid \\bar { \\mathbf { v } } \\right)$ . One could compute the $1 0 0 ( 1 - \\alpha ) \\%$ credible interval for $\\bar { \\theta _ { \\mathrm { c t f } } }$ using the following consistent estimators $\\pmb { \\| 3 9 \\| }$ : ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cb3c466dce116c46304e0686229449fa97000ce64fdab44d65288fe4321763bc.jpg",
+        "text": "$$\n\\hat { l } _ { \\alpha } ( T ) = \\theta ^ { ( \\lceil ( \\alpha / 2 ) T \\rceil ) } , \\hat { r } _ { \\alpha } ( T ) = \\theta ^ { ( \\lceil ( 1 - \\alpha / 2 ) T \\rceil ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            289,
+            758,
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+            779
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta ^ { ( \\lceil ( \\alpha / 2 ) T \\rceil ) } , \\theta ^ { ( \\lceil ( 1 - \\alpha / 2 ) T \\rceil ) }$ are the $\\lceil ( \\alpha / 2 ) T \\rceil$ th smallest and the $\\lceil ( 1 - \\alpha / 2 ) T \\rceil$ th smallest of $\\big \\{ \\theta ^ { ( t ) } \\big \\} ^ { 3 } .$ Our next results establish non-asymptotic deviation bounds for the empirical estimates of credible intervals defined in Eq. $\\textcircled { 1 6 }$ for finite samples. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. Fix 267 $T > 0$ and $\\delta \\in ( 0 , 1 )$ . Let function $f ( T , \\delta ) = \\sqrt { 2 T ^ { - 1 } \\ln ( 4 / \\delta ) }$ . With probability at least 268 $1 - \\delta$ , estimators $\\hat { l } _ { \\alpha } ( T ) , \\hat { r } _ { \\alpha } ( T )$ for any $\\alpha \\in [ 0 , 1 )$ is bounded by ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8cff452363ff02b12a055221888275764ce5726a969fea22ddd74679e90d8597.jpg",
+        "text": "$$\n\\hat { l } _ { \\alpha } ( T ) \\in \\left[ l _ { \\alpha - f ( T , \\delta ) } , l _ { \\alpha + f ( T , \\delta ) } \\right] , \\qquad \\hat { r } _ { \\alpha } ( T ) \\in \\left[ r _ { \\alpha + f ( T , \\delta ) } , r _ { \\alpha - f ( T , \\delta ) } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "3 For any real $\\alpha \\in \\mathbb { R }$ , $\\lceil \\alpha \\rceil$ denotes the smallest integer $n \\in \\mathbb { Z }$ larger than $\\alpha$ , i.e., $\\lceil \\alpha \\rceil = \\operatorname* { m i n } \\{ n \\in \\mathbb { Z } \\mid n \\geq \\alpha \\}$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "269 We summarize our algorithm, CREDIBLEIN  \n270 TERVAL, in Alg. $1 .$ It takes a credible level   \n271 $\\alpha$ and tolerance levels $\\delta , \\epsilon$ as inputs. In par  \n272 ticular, CREDIBLEINTERVAL repeatedly draw   \n273 $T \\ge \\lceil 2 \\epsilon ^ { - 2 } \\ln ( 4 / \\delta ) \\rceil$ samples from $P \\left( \\dot { \\theta _ { \\mathrm { c t f } } } \\mid \\bar { v } \\right)$ .   \n274 It then computes estimates $\\hat { l } _ { \\alpha } ( T ) , \\hat { h } _ { \\alpha } ( T )$ from   \n275 drawn samples following Eq. $\\textcircled { 1 6 }$ and return   \n276 them as the output. It follows immediately from   \n277 Lem. $\\bigstar$ that such a procedure efficiently approx  \n278 imates a $1 0 0 ( 1 - \\bar { \\alpha } ) \\%$ credible interval. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: CREDIBLEINTERVAL ",
+        "text_level": 1,
+        "bbox": [
+            498,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "1: Input: Credible level $\\alpha$ , tolerance level $\\delta , \\epsilon$ .   \n2: Output: An credible interval $[ l _ { \\alpha } , h _ { \\alpha } ]$ for $\\theta _ { \\mathrm { c t f } }$   \n3: Let $\\dot { T } = \\left. \\lceil 2 \\epsilon ^ { - 2 } \\ln ( 4 / \\delta ) \\right\\rceil$ .   \n4: Draw samples $\\big \\{ \\theta ^ { ( 1 ) } , \\dots , \\theta ^ { ( T ) } \\big \\}$ from the   \nposterior distribution $P \\left( \\theta _ { \\mathrm { c t f } } \\mid \\bar { \\mathbf { v } } \\right)$ .   \n5: Return interval $\\big [ \\hat { l } _ { \\alpha } ( T ) , \\hat { r } _ { \\alpha } ( T ) \\big ]$ (Eq. (16) ",
+        "bbox": [
+            503,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Corollary 3. Fix $\\delta \\in \\mathsf { \\Gamma } ( 0 , 1 )$ and $\\epsilon > 0$ . With probability at least $1 - \\delta$ , the interval $[ { \\hat { l } } , { \\hat { r } } ] =$ CREDIBLEINTERVAL $( \\alpha , \\delta , \\epsilon )$ for any $\\alpha \\in [ 0 , 1 )$ is bounded by $\\hat { l } \\in [ l _ { \\alpha - \\epsilon } , l _ { \\alpha + \\epsilon } ]$ and $\\hat { r } \\in [ r _ { \\alpha + \\epsilon } , r _ { \\alpha - \\epsilon } ]$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Corol. $3$ implies that any counterfactual parameter $\\theta _ { \\mathrm { c t f } }$ compatible with observational data $\\bar { \\mathbf { \\nabla } } \\bar { \\mathbf { v } }$ falls between $[ { \\hat { l } } , { \\hat { r } } ] =$ CREDIBLEINTERVAL $( \\alpha , \\delta , \\epsilon )$ with probability $P \\left( \\theta _ { \\mathrm { c t f } } \\in [ \\hat { l } , \\hat { r } ] \\mid \\bar { v } \\right) \\approx 1 - \\alpha \\pm \\epsilon$ . As the tolerance rate $\\epsilon  0 , [ \\hat { l } , \\hat { r } ]$ converges to a $1 0 0 ( 1 - \\alpha ) \\%$ credible interval with high probability. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "284 4 Simulations and Experiments ",
+        "text_level": 1,
+        "bbox": [
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+            359,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We demonstrate our algorithms on various simulated SCM instances and a real world patient dataset collected from the International Stroke Trial (IST) $\\mathbb { m }$ . Overall, we found that simulation results support our findings and the proposed bounding strategy consistently dominates state-of-art algorithms. When target distributions are identifiable (Experiment 1), our bounds collapse to the actual, unknown counterfactual probabilities. For non-identifiable settings, our algorithm obtains sharp asymptotic bounds when closed-form solutions already exist (Experiments 2 & 3); and improves over state-of-art bounds in other more general cases where the optimal strategy is unknown (Experiment 4). ",
+        "bbox": [
+            166,
+            393,
+            825,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In all experiments, we evaluate our proposed bounding strategy based on credible intervals $( c i )$ . In particular, we draw $4 \\times 1 0 ^ { 3 }$ samples from the posterior distribution over the target counterfactual $\\mathbf { \\bar { \\Psi } } ( \\theta _ { \\mathrm { c t f } } \\mid \\bar { V } )$ . This allows us to compute $1 0 0 \\%$ credible interval over $\\theta _ { \\mathrm { c t f } }$ within error $\\epsilon = 0 . 0 5$ , with probability at least $1 - \\delta = 0 . 9 5$ . As the baseline, we also include the actual counterfactual probability $\\theta ^ { * }$ . For details on simulation setups and additional experiments, we refer readers to Appendix C. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "297 Experiment 1: Frontdoor Graph This experiment evaluates our sam  \n298 pling algorithm on interventional probabilities that are identifiable from   \n299 the observational data. Consider the “Frontdoor” graph described in   \n300 Fig. $\\bigtriangledown$ where $X , Y , W$ are binary variables in $\\{ 0 , 1 \\}$ ; $U _ { 1 } , U _ { 2 } \\in \\mathbb { R }$ . In this   \n301 case, the interventional distribution $P ( y _ { x } )$ is identifiable from $P ( x , w , y )$   \n302 through the frontdoor adjustment $\\mathbb { \\left[ \\left. 3 3 \\right. \\right. }$ Thm. 3.3.4]. We collect $N = 1 0 ^ { 5 }$   \n303 observational samples $\\bar { \\cal V } = \\{ X ^ { ( n ) } , Y ^ { ( n ) } , W ^ { ( n ) } \\} _ { n = 1 } ^ { N }$ from a randomly ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/51afcc5bc58f94bd670f99dd40c8baf277d6e24ff5e68479eb604bfea3c9b594.jpg",
+        "image_caption": [
+            "Figure 3: Frontdoor "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            656,
+            588,
+            825,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "generated SCM. Fig. 4a shows samples drawn from the posterior distribution of the target probability $\\stackrel { \\prime } { P } ( Y _ { x = 0 } = 1 ) \\mid \\bar { V } )$ . The analysis reveals that these samples collapse to the actual interventional probability $P ( Y _ { x = 0 } = 1 ) = 0 . 5 0 8 5$ , which confirms the identifiability of $P ( y _ { x } )$ in Fig. 3. ",
+        "bbox": [
+            163,
+            683,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "307 Experiment 2: Instrumental Variables (IV) This experiment evaluates our bounding strategy in   \n308 non-identifiable settings, while closed-form solutions for the optimal bounds over target probabilities   \n309 already exist. Consider first the “IV” diagram in Fig. $^ { 1 \\mathrm { a } }$ where $X , Y , Z \\in \\{ 0 , 1 \\}$ and $U _ { 1 } , U _ { 2 } \\in \\mathbb { R }$   \n310 The non-identifiability of $P ( y _ { x } )$ from the observational data $P ( x , y , z )$ with the instrument $Z$ and the   \n311 unobserved confounding between $X$ and $Y$ has been acknowledged in [5]. For binary $X , Y , Z$ , [2]   \n312 derived closed-form, sharp bounds over $P ( y _ { x } )$ (labelled as opt). We collect $N = 1 0 ^ { 5 }$ observational   \n313 samples $\\bar { \\cal V } = \\{ X ^ { ( n ) } , Y ^ { ( n ) } , Z ^ { ( n ) } \\} _ { n = 1 } ^ { N }$ from a randomly generated SCM instance. Fig. $4  { \\mathbf { b } }$ shows   \n314 samples drawn from the posterior distribution of \u0000 $P ( Y _ { x = 0 } = 1 ) \\mid \\bar { V } )$ . As a baseline, we also include   \n315 the optimal bound opt, and posterior samples obtained from the Gibbs sampler of $\\mathbb { \\ m }$ , which utilizes   \n316 the canonical partitions of exogenous domains in $\\pmb { \\Vert 2 \\Vert }$ $( b p )$ . The analysis reveals that our algorithm   \n317 derives the valid bound over the actual probability $P ( Y _ { x = 0 } = 1 ) = 0 . 3 9 5 4$ ; the $1 0 0 \\%$ credible   \n318 interval converges to the optimal IV bound $l = 0 . 1 4 6 8 , r = 0 . 6 6 1 7$ . ",
+        "bbox": [
+            138,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/974221abcdde98c4222177846c82d05008fed3d1d6f770cc3c9c4ca44560435c.jpg",
+        "image_caption": [
+            "Figure 4: Histogram plots for samples drawn from the posterior distribution over target counterfactual probabilities. For all plots $( { \\bf a } - { \\bf d } )$ , ci represents our proposed algorithms; $b p$ stands for Gibbs samplers using the representation of canonical partitions $\\pmb { \\bigtriangledown } \\bar { \\bigtriangledown } \\bar { \\bigtriangledown }$ ; $\\theta ^ { * }$ is the actual counterfactual probability. $\\mathbf { \\underline { { \\sigma } } } ( \\mathbf { b } , \\mathbf { c } )$ opt represents the optimal asymptotic bound, if exists. (d) $n b$ stands for the natural bounds $\\pmb { \\mathbb { B } } \\pmb { \\mathbb { O } } \\Vert$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            87,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Experiment 3: Probability of Necessity and Sufficiency (PNS) We now study the problem of evaluating the probability of necessity and sufficiency $P ( Y _ { x = 1 } = 1 , Y _ { x = 0 } = 0 )$ from the observational data $P ( x , y )$ in the “Bow” diagram of Fig. $\\boxed { 1 \\dot { \\mathrm { d } } }$ where $X , Y \\in \\{ 0 , 1 \\}$ and $U \\in \\mathbb { R }$ . The sharp bound for $P ( Y _ { x = 1 } = 1 , Y _ { x = 0 } = 0 )$ from $P ( x , y )$ was introduced in $\\textcircled { \\lVert { 4 4 } \\rVert }$ (labelled as opt). We collect $N = 1 0 ^ { 5 }$ observational samples $\\bar { \\cal V } = \\{ X ^ { ( n ) } , Y ^ { ( n ) } \\} _ { n = 1 } ^ { N }$ from an SCM instance. Fig. $\\boxed { 4 \\mathrm { c } }$ shows samples drawn from the posterior distribution of $\\stackrel { \\prime } { P } ( Y _ { x = 1 } = 1 , Y _ { x = 0 } = 0 ) \\mid \\bar { V } )$ . As a baseline, we also include the optimal bound opt, and posterior samples obtained from the Gibbs sampler which discretizes the exogenous domains using canonical partitions $\\mathbb { I } \\mathbb { I } \\left( b p \\right)$ . The analysis reveals that our $1 0 0 \\%$ credible interval $( c i )$ matches the optimal PNS bound $l = 0 , r = 0 . 6 7 7 5$ , i.e., the proposed strategy achieves the sharp bound over the counterfactual probability $P ( Y _ { x = 1 } = 1 , Y _ { x = 0 } = 0 ) = 0 . 1 8 6 7$ . ",
+        "bbox": [
+            166,
+            309,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Experiment 4: International Stroke Trials (IST) IST was a large, randomized, open trial of up to 14 days of antithrombotic therapy after stroke onset $\\mathbb { \\ m }$ . In particular, the treatment $X$ is a pair $( i , j )$ where $i = 0$ stands for no aspirin allocation, 1 otherwise; $j = 0$ stands for no heparin allocation, 1 for median-dosage, and 2 for high-dosage. The primary outcome $Y \\in \\{ 0 , \\ldots , 3 \\}$ is the health of the patient 6 months after the treatment, where 0 stands for death, 1 for being dependent on the family, 2 for the partial recovery, and 3 for the full recovery. ",
+        "bbox": [
+            160,
+            467,
+            825,
+            551
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To emulate the presence of unobserved confounding, we filter the experimental data with selection rules $f _ { X } ^ { ( Z ) }$ , $Z \\in \\{ 0 , \\ldots , 9 \\}$ , following a procedure in $\\mathbb { H }$ . Doing so allows us to obtain $N = 3 \\times 1 0 ^ { 3 }$ synthetic observational samples $\\bar { \\cal V } = \\{ X ^ { ( n ) } , Y ^ { ( n ) } , Z ^ { ( n ) } \\} _ { n = 1 } ^ { N }$ that are compatible with the “Double bow” diagram of Fig. 1b. We are interested in evaluating the treatment effect $E [ Y _ { x = ( 1 , 0 ) } ]$ for only assigning aspirin $\\overline { { \\boldsymbol { X } } } = ( 1 , 0 )$ . Fig. 4d shows samples drawn from the posterior distribution of $\\left( E [ Y _ { x = ( 1 , 0 ) } ] \\mid \\bar { V } \\right)$ . As a baseline, we also include a naïve generalization of the discretization procedure $( b p ) \\pmb { \\mathbb { Z } }$ (see Appendix ${ \\bf D } )$ and the natural bounds $\\pm \\infty \\ldots$ estimated at the $9 5 \\%$ confidence level $( n b )$ [49]. Posterior samples of $c i$ and $b p$ are drawn using our proposed collapsed sampler due to the high-dimensional latent space. The analysis reveals that all algorithms achieve bounds that contain the actual, target causal effect $E [ Y _ { x = ( 1 , 0 ) } ] = 1 . 3 4 1 8$ . Our bounding strategy obtains a $1 0 0 \\%$ credible interval $l _ { c i } = 1 . 2 6 0 4 , r _ { c i } = 1 . 4 6 8 7$ , which consistently improves over all the other algorithms $( l _ { b p } = 1 . 1 1 2 1$ , $r _ { b p } = 1 . 8 0 7 3 , l _ { n b } = 1 . 1 1 9 5 , r _ { n b } = 1$ $r _ { n b } = 1 . 6 2 2 1$ ). ",
+        "bbox": [
+            165,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "47 5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            155,
+            755,
+            299,
+            772
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This paper investigated the problem of partial identification of counterfactual distributions, which concerns with bounding unknown counterfactual probabilities from the combination of the observational data and qualitative assumptions of the data-generating process, represented in the form of a directed acyclic causal diagram. We studied a special family of SCMs with discrete exogenous variables, taking values from a finite set of unobserved states, and showed that it could represent all counterfactual distributions (over finite observed variables) in an arbitrary causal diagram. That is, this new family of discrete SCMs is counterfactual equivalent to the original family of candidate SCMs compatible with the causal diagram. Using this result, we developed a novel algorithm to derive bounds over counterfactual probabilities from finite observations, which are provably tight. ",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] C. Avin, I. Shpitser, and J. Pearl. Identifiability of path-specific effects. In Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence IJCAI-05, pages 357–363, Edinburgh, UK, 2005. Morgan-Kaufmann Publishers.   \n[2] A. Balke and J. Pearl. Counterfactual probabilities: Computational methods, bounds, and applications. In R. L. de Mantaras and D. Poole, editors, Uncertainty in Artificial Intelligence 10, pages 46–54. Morgan Kaufmann, San Mateo, CA, 1994.   \n[3] A. Balke and J. Pearl. Counterfactuals and policy analysis in structural models. In P. Besnard and S. Hanks, editors, Uncertainty in Artificial Intelligence 11, pages 11–18. San Francisco, 1995.   \n[4] A. Balke and J. Pearl. Bounds on treatment effects from studies with imperfect compliance. Journal of the American Statistical Association, 92(439):1172–1176, September 1997.   \n[5] E. Bareinboim and J. Pearl. 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AAAI Press/The MIT Press, Menlo Park, CA, 2002.   \n[46] D. Todem, J. Fine, and L. Peng. A global sensitivity test for evaluating statistical hypotheses with nonidentifiable models. Biometrics, 66(2):558–566, 2010.   \n[47] S. Vansteelandt, E. Goetghebeur, M. G. Kenward, and G. Molenberghs. Ignorance and uncertainty regions as inferential tools in a sensitivity analysis. Statistica Sinica, pages 953–979, 2006.   \n[48] H. Waki, S. Kim, M. Kojima, and M. Muramatsu. Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity. SIAM Journal on Optimization, 17(1):218–242, 2006.   \n[49] J. Zhang and E. Bareinboim. Bounding causal effects on continuous outcomes. In Proceedings of the 35nd AAAI Conference on Artificial Intelligence, 2021. ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] “Throughout this paper, we assume that endogenous variables $V$ are discrete and finite; while exogenous variables $U$ could take any (continuous) value.”   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] This work does not present any foreseeable societal consequence.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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